Toss a Coin Ten Times and It Lies to You; Toss It a Thousand and It Confesses
Learn to place events on a scale from 0 to 1, calculate experimental probability from a table of trials, calculate theoretical probability by counting favourable outcomes, and see why the two values converge.
Why does a coin tossed ten times give a misleading answer?
Toss a fair coin ten times and count the heads. Getting is perfectly ordinary — it happens often. From those ten tosses the coin looks as though it favours heads per cent of the time.
Toss the same coin a thousand times and you might get heads. Now it looks almost exactly even.
Nothing about the coin changed. What changed is how much the count can wander relative to how many tosses there were. In ten tosses, being off by two is a huge fraction of ten. In a thousand tosses, being off by three is almost nothing.
So there are two different numbers to keep apart. The experimental probability is what actually happened:
The theoretical probability is what counting says should happen:
They are not the same thing, they rarely agree exactly, and they get closer as the trials pile up. This page covers the first part of the CBSE Class 9 Mathematics chapter on probability — the scale from to , both kinds of probability, and why they converge.
Toss the same coin a thousand times and you might get heads. Now it looks almost exactly even.
Nothing about the coin changed. What changed is how much the count can wander relative to how many tosses there were. In ten tosses, being off by two is a huge fraction of ten. In a thousand tosses, being off by three is almost nothing.
So there are two different numbers to keep apart. The experimental probability is what actually happened:
The theoretical probability is what counting says should happen:
They are not the same thing, they rarely agree exactly, and they get closer as the trials pile up. This page covers the first part of the CBSE Class 9 Mathematics chapter on probability — the scale from to , both kinds of probability, and why they converge.
How do you place an event on the scale from 0 to 1?
Ask how often the event would happen if the situation were repeated many times, then read off the scale, where means never and means always.
The five landmarks are:
- Impossible — . It cannot happen
- Unlikely — between and
- Equally likely — . As likely to happen as not
- Likely — between and
- Certain — . It must happen
Worked classifications, with a standard six-faced die.
- Rolling a — impossible, . No such face exists
- Rolling a — unlikely,
- Rolling an even number — equally likely,
- Rolling a number greater than — likely,
- Rolling a number less than — certain,
Worked classifications, in ordinary life.
- The sun rising tomorrow — certain
- A tossed coin landing on its edge and staying there — very nearly impossible
- Rain in Mumbai in a randomly chosen week of the monsoon — likely
- Drawing a red ball from a bag holding red and blue — equally likely
The complement rule. Since an event either happens or does not,
Worked example. , so .
**A probability can never exceed or fall below **, and an answer outside that range is a definite arithmetic error rather than an unusual result. A probability of or needs no interpreting; it needs correcting.
"Equally likely" describes the event, not a guess about it. A coin gives because its two outcomes are symmetric — there is no reason for either side to win. A drawing pin tossed in the air also lands two ways, point up or point down, and those are not equally likely, because the shape is not symmetric. **So has to be earned by an argument about the object**, and the counting formula in the third section applies only where that argument holds.
The five landmarks are:
- Impossible — . It cannot happen
- Unlikely — between and
- Equally likely — . As likely to happen as not
- Likely — between and
- Certain — . It must happen
Worked classifications, with a standard six-faced die.
- Rolling a — impossible, . No such face exists
- Rolling a — unlikely,
- Rolling an even number — equally likely,
- Rolling a number greater than — likely,
- Rolling a number less than — certain,
Worked classifications, in ordinary life.
- The sun rising tomorrow — certain
- A tossed coin landing on its edge and staying there — very nearly impossible
- Rain in Mumbai in a randomly chosen week of the monsoon — likely
- Drawing a red ball from a bag holding red and blue — equally likely
The complement rule. Since an event either happens or does not,
Worked example. , so .
**A probability can never exceed or fall below **, and an answer outside that range is a definite arithmetic error rather than an unusual result. A probability of or needs no interpreting; it needs correcting.
"Equally likely" describes the event, not a guess about it. A coin gives because its two outcomes are symmetric — there is no reason for either side to win. A drawing pin tossed in the air also lands two ways, point up or point down, and those are not equally likely, because the shape is not symmetric. **So has to be earned by an argument about the object**, and the counting formula in the third section applies only where that argument holds.
Formula
How do you calculate probability from a table of trial results?
Divide the count for the outcome by the total number of trials. That is the experimental or empirical probability:
Worked example 1 — coin tosses. A coin is tossed times and lands heads times:
The tails count must be , giving . Check: .
Worked example 2 — a die rolled 60 times. The results are recorded as:
- appeared times
- appeared times
- appeared times
- appeared times
- appeared times
- appeared times
First check the total: . Correct, so no reading was lost.
Worked example 3 — a survey. Of households surveyed, own a bicycle. Then
and for a household picked at random from that group, the probability of not owning one is .
Worked example 4 — germinating seeds. Out of seeds sown, germinate:
So out of seeds from the same batch, roughly would be expected to germinate.
**The probabilities of all the outcomes must add to exactly .** In worked example 2 the six values are , and their numerators total . That is the single best check on a table question, and it catches a miscounted row instantly.
Experimental probability is a measurement, so it depends on the trials. Roll the same die another times and every value will shift. It is a report about what happened, not a property of the die — which is exactly the distinction the next section draws.
Worked example 1 — coin tosses. A coin is tossed times and lands heads times:
The tails count must be , giving . Check: .
Worked example 2 — a die rolled 60 times. The results are recorded as:
- appeared times
- appeared times
- appeared times
- appeared times
- appeared times
- appeared times
First check the total: . Correct, so no reading was lost.
Worked example 3 — a survey. Of households surveyed, own a bicycle. Then
and for a household picked at random from that group, the probability of not owning one is .
Worked example 4 — germinating seeds. Out of seeds sown, germinate:
So out of seeds from the same batch, roughly would be expected to germinate.
**The probabilities of all the outcomes must add to exactly .** In worked example 2 the six values are , and their numerators total . That is the single best check on a table question, and it catches a miscounted row instantly.
Experimental probability is a measurement, so it depends on the trials. Roll the same die another times and every value will shift. It is a report about what happened, not a property of the die — which is exactly the distinction the next section draws.
How do you count the theoretical probability instead?
Count the outcomes that give the event, count all the outcomes, and divide — provided every outcome is equally likely:
No experiment is performed. The answer comes from the structure of the situation.
Worked example 1 — a die. There are equally likely faces.
-
- , since , , are favourable
- , from and
- , from and
- and
Worked example 2 — a bag of balls. A bag holds red, blue and green balls, so in total:
Check: . Correct.
And , which also equals — the complement rule and direct counting agreeing.
Worked example 3 — a pack of cards. A standard pack has cards, in each of four suits, with cards of each rank.
-
-
-
- , counting jack, queen and king in each suit
-
Worked example 4 — two-digit numbers. A number is picked at random from to , giving possibilities. How many are multiples of ? They are , which is numbers:
The formula demands equally likely outcomes, and that is a real condition. For the bag it holds because the balls are identical apart from colour. It would fail if the green balls were larger and easier to grab, and it fails for a bent die.
A common misuse. Tossing two coins, the outcomes are sometimes listed as two heads, one head, no heads — three cases, suggesting . But one head can happen two ways, head-then-tail and tail-then-head, so the three cases are not equally likely and the true answer is . Listing the outcomes properly is the whole job, and the next part of this chapter is about doing that systematically.
No experiment is performed. The answer comes from the structure of the situation.
Worked example 1 — a die. There are equally likely faces.
-
- , since , , are favourable
- , from and
- , from and
- and
Worked example 2 — a bag of balls. A bag holds red, blue and green balls, so in total:
Check: . Correct.
And , which also equals — the complement rule and direct counting agreeing.
Worked example 3 — a pack of cards. A standard pack has cards, in each of four suits, with cards of each rank.
-
-
-
- , counting jack, queen and king in each suit
-
Worked example 4 — two-digit numbers. A number is picked at random from to , giving possibilities. How many are multiples of ? They are , which is numbers:
The formula demands equally likely outcomes, and that is a real condition. For the bag it holds because the balls are identical apart from colour. It would fail if the green balls were larger and easier to grab, and it fails for a bent die.
A common misuse. Tossing two coins, the outcomes are sometimes listed as two heads, one head, no heads — three cases, suggesting . But one head can happen two ways, head-then-tail and tail-then-head, so the three cases are not equally likely and the true answer is . Listing the outcomes properly is the whole job, and the next part of this chapter is about doing that systematically.
Why do experimental and theoretical probability move closer together?
Because the wandering of the count grows more slowly than the number of trials, so the fraction settles even though the count keeps drifting.
A worked sequence of coin experiments. The theoretical value is .
- tosses, heads: experimental , off by
- tosses, heads: , off by
- tosses, heads: , off by
- tosses, heads: , off by
- tosses, heads: , off by
The gap shrinks steadily. This is the law of large numbers, and it is the reason theoretical probability is useful for predicting real situations at all.
Now count the same data differently. How far is the head count from exactly half?
- tosses: half is , so is ** away**
- tosses: half is , so is ** away**
- tosses: half is , so is ** away
The count gap did not shrink — it grew. What shrank is the gap as a fraction of the total**, because is far smaller than even though .
That is the point most often missed. More trials do not push the head count towards exactly half; they make the proportion reliable while the raw difference drifts further from zero. Convergence is a statement about the ratio, not about the counts.
Worked example — using convergence to predict. A batch of seeds germinated times, giving . For a new consignment of seeds from the same batch, roughly
would be expected to germinate. This prediction is only as good as the batch being similar, which is the assumption doing all the work.
The coin has no memory. After five heads in a row, the next toss is still — the coin carries no record of the previous tosses and nothing is owed to tails. The belief that a run must be corrected is the commonest error about probability, and convergence does not support it: the proportion settles because the totals grow, not because later results compensate for earlier ones.
When the two values stay apart, suspect the assumption, not the mathematics. A die rolled times giving sixes has an experimental probability of against a theoretical . With that many trials the gap is too large to shrug off, and the sensible conclusion is that the die is not fair — so the theoretical value was computed for the wrong object.
A worked sequence of coin experiments. The theoretical value is .
- tosses, heads: experimental , off by
- tosses, heads: , off by
- tosses, heads: , off by
- tosses, heads: , off by
- tosses, heads: , off by
The gap shrinks steadily. This is the law of large numbers, and it is the reason theoretical probability is useful for predicting real situations at all.
Now count the same data differently. How far is the head count from exactly half?
- tosses: half is , so is ** away**
- tosses: half is , so is ** away**
- tosses: half is , so is ** away
The count gap did not shrink — it grew. What shrank is the gap as a fraction of the total**, because is far smaller than even though .
That is the point most often missed. More trials do not push the head count towards exactly half; they make the proportion reliable while the raw difference drifts further from zero. Convergence is a statement about the ratio, not about the counts.
Worked example — using convergence to predict. A batch of seeds germinated times, giving . For a new consignment of seeds from the same batch, roughly
would be expected to germinate. This prediction is only as good as the batch being similar, which is the assumption doing all the work.
The coin has no memory. After five heads in a row, the next toss is still — the coin carries no record of the previous tosses and nothing is owed to tails. The belief that a run must be corrected is the commonest error about probability, and convergence does not support it: the proportion settles because the totals grow, not because later results compensate for earlier ones.
When the two values stay apart, suspect the assumption, not the mathematics. A die rolled times giving sixes has an experimental probability of against a theoretical . With that many trials the gap is too large to shrug off, and the sensible conclusion is that the die is not fair — so the theoretical value was computed for the wrong object.
Exam tip
Exam tip: check the outcome counts add up before you divide
Add the frequency column and check it matches the stated total. In a -roll table, confirms nothing was lost.
**Check your probabilities sum to across all outcomes. It is the fastest way to catch a miscount.
Give the answer as a fraction in lowest terms** unless a decimal is asked for: .
**Never write a probability outside to .** A value of or is an error, not a finding.
Use the complement for "not" and "at least": .
State the total number of outcomes explicitly before dividing — cards, balls, two-digit numbers. Most lost marks come from a wrong denominator.
Say which kind of probability you used. Write experimental when the data came from trials and theoretical when it came from counting; questions ask for a specific one.
The counting formula needs equally likely outcomes — say so when you use it, and note when it fails (a bent die, a drawing pin).
Two coins have FOUR outcomes, not three: HH, HT, TH, TT.
For a prediction, multiply the probability by the new total: seeds.
And remember the coin has no memory — a run of heads does not raise the chance of a tail.
**Check your probabilities sum to across all outcomes. It is the fastest way to catch a miscount.
Give the answer as a fraction in lowest terms** unless a decimal is asked for: .
**Never write a probability outside to .** A value of or is an error, not a finding.
Use the complement for "not" and "at least": .
State the total number of outcomes explicitly before dividing — cards, balls, two-digit numbers. Most lost marks come from a wrong denominator.
Say which kind of probability you used. Write experimental when the data came from trials and theoretical when it came from counting; questions ask for a specific one.
The counting formula needs equally likely outcomes — say so when you use it, and note when it fails (a bent die, a drawing pin).
Two coins have FOUR outcomes, not three: HH, HT, TH, TT.
For a prediction, multiply the probability by the new total: seeds.
And remember the coin has no memory — a run of heads does not raise the chance of a tail.
Did you know
Why more tosses make the fraction steadier and the count wilder
Here is a fact that feels wrong the first several times you meet it. As you toss a coin more and more, the proportion of heads settles down towards a half, while the number of heads wanders further and further from exactly half.
Both things are happening at once, and the coin experiments in the last section show them side by side. At tosses the head count was away from half; at tosses it was away. Meanwhile the proportion went from being off to being off.
The reason is a difference in growth rates. The wandering of the count grows roughly like the square root of the number of tosses, while the number of tosses grows like the number of tosses. So dividing one by the other leaves something that shrinks — and the fraction settles precisely because the denominator outruns the numerator's drift.
A rough sense of scale makes it concrete. In tosses, being or so away from is unremarkable. In tosses, being away from is equally unremarkable — ten times the drift, but a hundred times the tosses, so the proportion is ten times closer to a half.
This is why the phrase the law of averages is misleading if it is taken to mean the counts even out. They do not. The ratios even out, and only because the totals grow faster than the errors do.
It also explains why a casino or an insurer can be confident about many customers and not about one. A single outcome is unpredictable and stays unpredictable. Ten thousand of them have a proportion you can rely on — and the same argument that makes the proportion reliable also guarantees that the raw surplus of wins or losses will keep growing.
Both things are happening at once, and the coin experiments in the last section show them side by side. At tosses the head count was away from half; at tosses it was away. Meanwhile the proportion went from being off to being off.
The reason is a difference in growth rates. The wandering of the count grows roughly like the square root of the number of tosses, while the number of tosses grows like the number of tosses. So dividing one by the other leaves something that shrinks — and the fraction settles precisely because the denominator outruns the numerator's drift.
A rough sense of scale makes it concrete. In tosses, being or so away from is unremarkable. In tosses, being away from is equally unremarkable — ten times the drift, but a hundred times the tosses, so the proportion is ten times closer to a half.
This is why the phrase the law of averages is misleading if it is taken to mean the counts even out. They do not. The ratios even out, and only because the totals grow faster than the errors do.
It also explains why a casino or an insurer can be confident about many customers and not about one. A single outcome is unpredictable and stays unpredictable. Ten thousand of them have a proportion you can rely on — and the same argument that makes the proportion reliable also guarantees that the raw surplus of wins or losses will keep growing.
Exam relevance
How does probability feed into JEE Main and NEET?
Because the counting definition established here is the base of an entire Class 11 and 12 chapter, and the convergence idea underpins how experimental science reports anything at all.
This is the foundation for Class 11 Mathematics Probability and Class 12 Probability, both examined in JEE Main. The definition
is carried forward unchanged; what grows is the difficulty of counting and . Class 11 Permutations and Combinations exists largely to make those counts possible for situations too big to list, and every card and ball problem there is this page's problem with doing the counting.
The complement rule is the most reused single line. becomes the standard first move for any question containing at least one, and it stays the standard move through Class 12 where the alternative is a long sum. A student who reaches for the complement automatically saves time on a recurring JEE Main question type.
The equally-likely condition is what later theory replaces. Class 12 introduces conditional probability and Bayes' theorem precisely because most real situations do not have equally likely outcomes, and the axiomatic definition of probability generalises the counting formula so it still applies. The drawing-pin example on this page is the reason that generalisation is needed.
Where the experimental side goes. Class 11 Statistics treats the spread of data, and the observation here — that a proportion settles while a count drifts — is the qualitative form of the standard-deviation result. For NEET Biology, the ratios in genetics are exactly this material: a phenotype ratio is a theoretical probability of , and the fact that a real cross gives and rather than and is the law of large numbers at work. Questions on expected offspring numbers are probability questions in biological clothing.
What the questions look like. For board work, expect classify events on the scale, experimental probability from a frequency table, theoretical probability from a die, a bag or a pack of cards, a prediction for a larger batch, and one comparison question on why the two values differ. For JEE Main, probability appears with combinations doing the counting, and almost always needs the complement or a careful sample space.
How board and competitive emphasis differ. A board paper rewards showing the denominator and the frequency check. A competitive paper assumes both and tests the counting — whether you can enumerate the favourable cases without listing them.
The single trap that costs the most marks. Using an unequal sample space. Two coins give four outcomes, not three, and treating one head as a single case gives instead of . The error is invisible in the final answer, since both values are legitimate-looking probabilities. The defence is to list the sample space in full for any small experiment before dividing anything — which is exactly what the next part of this chapter builds into a method.
This is the foundation for Class 11 Mathematics Probability and Class 12 Probability, both examined in JEE Main. The definition
is carried forward unchanged; what grows is the difficulty of counting and . Class 11 Permutations and Combinations exists largely to make those counts possible for situations too big to list, and every card and ball problem there is this page's problem with doing the counting.
The complement rule is the most reused single line. becomes the standard first move for any question containing at least one, and it stays the standard move through Class 12 where the alternative is a long sum. A student who reaches for the complement automatically saves time on a recurring JEE Main question type.
The equally-likely condition is what later theory replaces. Class 12 introduces conditional probability and Bayes' theorem precisely because most real situations do not have equally likely outcomes, and the axiomatic definition of probability generalises the counting formula so it still applies. The drawing-pin example on this page is the reason that generalisation is needed.
Where the experimental side goes. Class 11 Statistics treats the spread of data, and the observation here — that a proportion settles while a count drifts — is the qualitative form of the standard-deviation result. For NEET Biology, the ratios in genetics are exactly this material: a phenotype ratio is a theoretical probability of , and the fact that a real cross gives and rather than and is the law of large numbers at work. Questions on expected offspring numbers are probability questions in biological clothing.
What the questions look like. For board work, expect classify events on the scale, experimental probability from a frequency table, theoretical probability from a die, a bag or a pack of cards, a prediction for a larger batch, and one comparison question on why the two values differ. For JEE Main, probability appears with combinations doing the counting, and almost always needs the complement or a careful sample space.
How board and competitive emphasis differ. A board paper rewards showing the denominator and the frequency check. A competitive paper assumes both and tests the counting — whether you can enumerate the favourable cases without listing them.
The single trap that costs the most marks. Using an unequal sample space. Two coins give four outcomes, not three, and treating one head as a single case gives instead of . The error is invisible in the final answer, since both values are legitimate-looking probabilities. The defence is to list the sample space in full for any small experiment before dividing anything — which is exactly what the next part of this chapter builds into a method.
Key takeaways
The probability scale, experimental and theoretical values: quick revision
- **Every probability satisfies . A value outside that range is an arithmetic error.
- Impossible** is , certain is , equally likely is ; between them lie unlikely and likely.
- On a die: , , , , .
- Complement rule: , so .
- Experimental probability — a measurement, so it changes with the trials.
- heads in tosses gives , and tails gives ; the two add to .
- A die rolled times with counts : the total is , , , .
- Add the frequency column first — it is the best check on a table question.
- of households gives , so ; of seeds gives .
- Theoretical probability , and needs equally likely outcomes.
- Bag of red, blue, green: , , , summing to .
- Pack of : , , , , .
- From to there are numbers, and multiples of , so .
- The equally-likely condition can fail — a bent die or a drawing pin does not give by symmetry.
- Two coins have four outcomes, so , not .
- Convergence: , then , , , — the gap from falls to .
- But the count gap grows: away at tosses, away at . Convergence is about the ratio, not the counts.
- To predict, multiply: seeds expected to germinate.
- The coin has no memory — after five heads the next toss is still .
- A large persistent gap means the object is not fair, not that the mathematics failed: sixes in rolls gives against .
Toss a coin twenty times, write the running proportion of heads after every toss, and watch how violently it swings early and how little it moves by the end.
- Impossible** is , certain is , equally likely is ; between them lie unlikely and likely.
- On a die: , , , , .
- Complement rule: , so .
- Experimental probability — a measurement, so it changes with the trials.
- heads in tosses gives , and tails gives ; the two add to .
- A die rolled times with counts : the total is , , , .
- Add the frequency column first — it is the best check on a table question.
- of households gives , so ; of seeds gives .
- Theoretical probability , and needs equally likely outcomes.
- Bag of red, blue, green: , , , summing to .
- Pack of : , , , , .
- From to there are numbers, and multiples of , so .
- The equally-likely condition can fail — a bent die or a drawing pin does not give by symmetry.
- Two coins have four outcomes, so , not .
- Convergence: , then , , , — the gap from falls to .
- But the count gap grows: away at tosses, away at . Convergence is about the ratio, not the counts.
- To predict, multiply: seeds expected to germinate.
- The coin has no memory — after five heads the next toss is still .
- A large persistent gap means the object is not fair, not that the mathematics failed: sixes in rolls gives against .
Toss a coin twenty times, write the running proportion of heads after every toss, and watch how violently it swings early and how little it moves by the end.