Two Coins Have Four Outcomes, Not Three, and It Matters
Learn to list the sample space of an experiment, calculate probability as a ratio of outcomes, handle coins, dice, cards and balls, and use the complement rule.
Is the chance of getting exactly one head on two coins one third?
No — it is one half, and the reason is the most important idea in the chapter.
Tossing two coins, it is tempting to say there are three results: two heads, one head, or no heads. Three results, so each has probability .
But one head can happen in two distinct ways. The first coin could show a head and the second a tail, or the other way round. Writing the outcomes out properly:
There are four equally likely outcomes, and two of them have exactly one head, so the probability is .
The error was listing three results that are not equally likely and treating them as though they were. Every probability calculation in this chapter rests on the outcomes being equally likely, and listing the sample space properly is how you make sure they are. This page covers the ICSE Class 8 Mathematics chapter on probability.
Tossing two coins, it is tempting to say there are three results: two heads, one head, or no heads. Three results, so each has probability .
But one head can happen in two distinct ways. The first coin could show a head and the second a tail, or the other way round. Writing the outcomes out properly:
There are four equally likely outcomes, and two of them have exactly one head, so the probability is .
The error was listing three results that are not equally likely and treating them as though they were. Every probability calculation in this chapter rests on the outcomes being equally likely, and listing the sample space properly is how you make sure they are. This page covers the ICSE Class 8 Mathematics chapter on probability.
What are an experiment, an outcome, a sample space and an event?
Four words that must be used precisely, because the formula depends on them.
- A random experiment is an action whose result cannot be predicted in advance, though all the possible results are known — tossing a coin, rolling a die, drawing a card.
- An outcome is one possible result of the experiment.
- The sample space is the set of all possible outcomes. The number of them is written .
- An event is any collection of outcomes that the question is interested in, written , with outcomes in it.
Worked example 1 — one coin. , so . The event getting a head is with .
Worked example 2 — one die. , so . The event getting an even number is with .
Worked example 3 — three coins. Listing carefully:
The event exactly two heads is , so .
How many outcomes to expect. One coin gives , two coins give , three coins give — each extra coin doubles the count, because each of the earlier outcomes can be followed by a head or a tail. So coins give outcomes, and knowing the total in advance tells you when your list is complete. Similarly two dice give .
List the sample space in a system, not at random. For three coins, write all the outcomes starting with first, then all starting with . A haphazard list is how outcomes get missed or repeated, and tells you exactly how many to find.
Two coins tossed together and one coin tossed twice have the same sample space. Both give , because in each case there are two independent results to record. Questions use both wordings for the same mathematics, so neither should surprise you.
- A random experiment is an action whose result cannot be predicted in advance, though all the possible results are known — tossing a coin, rolling a die, drawing a card.
- An outcome is one possible result of the experiment.
- The sample space is the set of all possible outcomes. The number of them is written .
- An event is any collection of outcomes that the question is interested in, written , with outcomes in it.
Worked example 1 — one coin. , so . The event getting a head is with .
Worked example 2 — one die. , so . The event getting an even number is with .
Worked example 3 — three coins. Listing carefully:
The event exactly two heads is , so .
How many outcomes to expect. One coin gives , two coins give , three coins give — each extra coin doubles the count, because each of the earlier outcomes can be followed by a head or a tail. So coins give outcomes, and knowing the total in advance tells you when your list is complete. Similarly two dice give .
List the sample space in a system, not at random. For three coins, write all the outcomes starting with first, then all starting with . A haphazard list is how outcomes get missed or repeated, and tells you exactly how many to find.
Two coins tossed together and one coin tossed twice have the same sample space. Both give , because in each case there are two independent results to record. Questions use both wordings for the same mathematics, so neither should surprise you.
Formula
What is the formula for the probability of an event?
For an experiment whose outcomes are equally likely,
Worked example 1. The probability of a head on one coin is ; of an even number on a die, .
Worked example 2 — one die, several events. With :
- Prime number: , so
- **Multiple of **: , so
- **Greater than **: , so
- **Less than **: no outcomes, so
- **At most **: all six outcomes, so
The range of every probability.
An event with is impossible; one with is sure or certain. A probability can never exceed or fall below , so an answer of or is not an unusual result but a mistake — and that check takes a second.
Read the comparison words exactly. *Less than is impossible on a die, while less than * has probability — the single outcome . And **at most ** means with , while **less than ** means with . The wording is worth as much as the arithmetic here.
Probability may be written as a fraction, a decimal or a percentage. So , and all say the same thing. Leave it as a fraction in lowest terms unless the question asks otherwise, since that is exact and a rounded decimal is not.
Equally likely is an assumption, not a fact. The formula needs a fair coin and an unbiased die. A weighted die still has six outcomes, but they are no longer equally likely and no longer applies. That is why examination questions say a fair coin or an unbiased die — it is the condition that makes the whole method valid.
Worked example 1. The probability of a head on one coin is ; of an even number on a die, .
Worked example 2 — one die, several events. With :
- Prime number: , so
- **Multiple of **: , so
- **Greater than **: , so
- **Less than **: no outcomes, so
- **At most **: all six outcomes, so
The range of every probability.
An event with is impossible; one with is sure or certain. A probability can never exceed or fall below , so an answer of or is not an unusual result but a mistake — and that check takes a second.
Read the comparison words exactly. *Less than is impossible on a die, while less than * has probability — the single outcome . And **at most ** means with , while **less than ** means with . The wording is worth as much as the arithmetic here.
Probability may be written as a fraction, a decimal or a percentage. So , and all say the same thing. Leave it as a fraction in lowest terms unless the question asks otherwise, since that is exact and a rounded decimal is not.
Equally likely is an assumption, not a fact. The formula needs a fair coin and an unbiased die. A weighted die still has six outcomes, but they are no longer equally likely and no longer applies. That is why examination questions say a fair coin or an unbiased die — it is the condition that makes the whole method valid.
How do you find probabilities for coins, dice, cards and balls?
**Count first, then count the favourable outcomes carefully.
Worked example 1 — two coins.** With and :
- Exactly one head: , so
- At least one head: , so
- No head: , so
- Two heads: , so
Worked example 2 — three coins. With :
- Exactly two heads:
- At least two heads: , so
At least differs from exactly. At least two heads includes the case of three, while exactly two excludes it — which is why the two answers above are and .
Worked example 3 — two dice. Rolling two dice gives ordered pairs.
- **Sum **: — six ways, so
- **Sum **: — five ways, so
- A doublet (both dice the same): six ways, so
- **Sum **: only , so
- **Sum greater than **: — three ways, so
- **Sum **: impossible, so
The two dice are distinguishable, even if identical. and are different outcomes — first die and second , against first and second . Counting them as one gives instead of and every answer comes out wrong. It is the same error as the two-coin mistake in the opening section, in a bigger setting.
Worked example 4 — a pack of cards. A standard pack has cards: suits of , with hearts and diamonds red, spades and clubs black. The face cards are the jack, queen and king of each suit, so there are .
- A king:
- A heart:
- A red card:
- A face card:
- The ace of spades:
- A red king: only the king of hearts and the king of diamonds, so
Worked example 5 — balls in a bag. A bag holds red, blue and green balls, so .
Check by adding them: . When the events cover every possibility and do not overlap, their probabilities must total — and that is a complete check on a whole question.
Worked example 6. A bag has white and black balls. Then and , totalling as required.
Worked example 1 — two coins.** With and :
- Exactly one head: , so
- At least one head: , so
- No head: , so
- Two heads: , so
Worked example 2 — three coins. With :
- Exactly two heads:
- At least two heads: , so
At least differs from exactly. At least two heads includes the case of three, while exactly two excludes it — which is why the two answers above are and .
Worked example 3 — two dice. Rolling two dice gives ordered pairs.
- **Sum **: — six ways, so
- **Sum **: — five ways, so
- A doublet (both dice the same): six ways, so
- **Sum **: only , so
- **Sum greater than **: — three ways, so
- **Sum **: impossible, so
The two dice are distinguishable, even if identical. and are different outcomes — first die and second , against first and second . Counting them as one gives instead of and every answer comes out wrong. It is the same error as the two-coin mistake in the opening section, in a bigger setting.
Worked example 4 — a pack of cards. A standard pack has cards: suits of , with hearts and diamonds red, spades and clubs black. The face cards are the jack, queen and king of each suit, so there are .
- A king:
- A heart:
- A red card:
- A face card:
- The ace of spades:
- A red king: only the king of hearts and the king of diamonds, so
Worked example 5 — balls in a bag. A bag holds red, blue and green balls, so .
Check by adding them: . When the events cover every possibility and do not overlap, their probabilities must total — and that is a complete check on a whole question.
Worked example 6. A bag has white and black balls. Then and , totalling as required.
How does the complement rule save you work?
**The probability of an event and of it not happening must add to :**
This is true because every outcome either belongs to or does not, so .
Worked example 1. If , then .
Worked example 2. From the bag of red, blue and green balls:
Check directly: the non-red balls are the blue and green, five in all, so . The two routes agree.
Worked example 3 — where the rule genuinely saves effort. Two dice are rolled. What is the probability that the sum is not ?
Counting directly would mean listing thirty outcomes. Using the complement:
One subtraction instead of thirty entries.
Worked example 4 — with three coins. The probability of at least one head on three coins. The complement of at least one head is no heads at all, which is the single outcome :
**Look for the complement whenever a question says at least or not. Those two phrasings are the signal, and the complement is nearly always the shorter count.
Worked example 5 — sure and impossible events.** Rolling one die:
- *Getting a number less than * is a sure event,
- *Getting * is an impossible event,
And they are complements of each other: , as the rule requires.
Worked example 6 — finding a count from a probability. A bag contains red balls and some blue ones. If the probability of drawing a blue ball is , how many blue balls are there?
Let there be blue balls, so the total is :
Check: with blue and red, the total is and . Correct — and note that , which is indeed .
The complement is not the opposite in colour but in logic. For the three-colour bag, the complement of red is blue or green — everything else, not merely the next colour listed. Getting that wrong turns a one-line answer into a wrong one, so name the complement in words before computing it.
This is true because every outcome either belongs to or does not, so .
Worked example 1. If , then .
Worked example 2. From the bag of red, blue and green balls:
Check directly: the non-red balls are the blue and green, five in all, so . The two routes agree.
Worked example 3 — where the rule genuinely saves effort. Two dice are rolled. What is the probability that the sum is not ?
Counting directly would mean listing thirty outcomes. Using the complement:
One subtraction instead of thirty entries.
Worked example 4 — with three coins. The probability of at least one head on three coins. The complement of at least one head is no heads at all, which is the single outcome :
**Look for the complement whenever a question says at least or not. Those two phrasings are the signal, and the complement is nearly always the shorter count.
Worked example 5 — sure and impossible events.** Rolling one die:
- *Getting a number less than * is a sure event,
- *Getting * is an impossible event,
And they are complements of each other: , as the rule requires.
Worked example 6 — finding a count from a probability. A bag contains red balls and some blue ones. If the probability of drawing a blue ball is , how many blue balls are there?
Let there be blue balls, so the total is :
Check: with blue and red, the total is and . Correct — and note that , which is indeed .
The complement is not the opposite in colour but in logic. For the three-colour bag, the complement of red is blue or green — everything else, not merely the next colour listed. Getting that wrong turns a one-line answer into a wrong one, so name the complement in words before computing it.
Exam tip
Exam tip: list the sample space, and check the probabilities total one
Write out the sample space before calculating. For two coins it is — four outcomes, not three, so exactly one head has probability .
**Know in advance**: coins give , two dice give , a pack gives , a bag gives the total number of balls. Then you know when your list is complete.
Order matters with two dice. and are different outcomes.
List systematically — all the outcomes, then all the ones.
Read the wording exactly: at least two heads is on three coins while exactly two is ; *at most * on a die is while *less than * is .
**Use the complement for at least and not**: , and .
Name the complement in words first — the complement of red in a three-colour bag is blue or green.
**Check that , and that a full set of non-overlapping events totals exactly .
Leave answers as fractions in lowest terms, and remember the formula needs a fair coin or unbiased** die.
**Know in advance**: coins give , two dice give , a pack gives , a bag gives the total number of balls. Then you know when your list is complete.
Order matters with two dice. and are different outcomes.
List systematically — all the outcomes, then all the ones.
Read the wording exactly: at least two heads is on three coins while exactly two is ; *at most * on a die is while *less than * is .
**Use the complement for at least and not**: , and .
Name the complement in words first — the complement of red in a three-colour bag is blue or green.
**Check that , and that a full set of non-overlapping events totals exactly .
Leave answers as fractions in lowest terms, and remember the formula needs a fair coin or unbiased** die.
Did you know
Why a fair coin does not owe you a head
Toss a fair coin five times and get five tails. What is the probability of a head on the sixth toss?
It is . Exactly what it was on the first toss, and on every toss before it.
The coin has no memory. It is a piece of metal with two faces, and it carries no record of what it did a moment ago, nor any obligation to even things up. The feeling that a head is due is one of the most persistent misunderstandings about probability, and the sample space explains why it is wrong: the experiment toss the coin once has whatever happened previously, so every time.
What is true is that over very many tosses the fraction of heads tends to settle near . That is a statement about long runs, not about the next toss, and the difference between the two is the whole of the confusion. Five tails in a row are not corrected by a head; they are simply diluted by the hundreds of tosses that follow.
There is a related trap worth noticing. The probability of five tails in a row before you start is , which is genuinely unlikely. But once four tails have already happened, the probability of the fifth being a tail is back to — the unlikely part was the whole sequence, not the last step.
So the theoretical probability in this chapter answers what fraction of the equally likely outcomes are favourable? It never answers what is about to happen?, and no amount of past results changes it.
It is . Exactly what it was on the first toss, and on every toss before it.
The coin has no memory. It is a piece of metal with two faces, and it carries no record of what it did a moment ago, nor any obligation to even things up. The feeling that a head is due is one of the most persistent misunderstandings about probability, and the sample space explains why it is wrong: the experiment toss the coin once has whatever happened previously, so every time.
What is true is that over very many tosses the fraction of heads tends to settle near . That is a statement about long runs, not about the next toss, and the difference between the two is the whole of the confusion. Five tails in a row are not corrected by a head; they are simply diluted by the hundreds of tosses that follow.
There is a related trap worth noticing. The probability of five tails in a row before you start is , which is genuinely unlikely. But once four tails have already happened, the probability of the fifth being a tail is back to — the unlikely part was the whole sequence, not the last step.
So the theoretical probability in this chapter answers what fraction of the equally likely outcomes are favourable? It never answers what is about to happen?, and no amount of past results changes it.
Key takeaways
Probability: quick revision
- A random experiment has unpredictable results; an outcome is one result; the sample space is all of them, with outcomes; an event is a collection of outcomes.
- , valid only when the outcomes are equally likely — so a fair coin and an unbiased die.
- . An impossible event has , a sure event has . An answer outside that range is a mistake.
- Know the totals: coins give outcomes, two dice give , a pack gives .
- Two coins give — four outcomes. Exactly one head is , at least one head , no head , two heads .
- Three coins give outcomes: exactly two heads is , at least two heads .
- One die: even , prime , multiple of is , greater than is . *At most * is but *less than * is .
- Two dice: sum is , sum is , a doublet , sum is , sum above is , sum is .
- Order matters: and are different outcomes, which is why and not .
- Cards: a king , a heart , a red card , a face card , the ace of spades , a red king .
- Balls: red, blue, green give , and , which total . And white with black give and .
- Complement rule: . So gives ; ; .
- **Use the complement whenever you see at least or not**, and name it in words first — the complement of red is blue or green.
- Finding a count: red with gives , so .
- A set of non-overlapping events covering everything must total **exactly .
- A fair coin has no memory** — five tails do not make a head more likely next time.
Write out all outcomes for two dice once, by hand, and count how many give each sum from to — the shape that appears is worth more than any formula you could memorise instead.
- , valid only when the outcomes are equally likely — so a fair coin and an unbiased die.
- . An impossible event has , a sure event has . An answer outside that range is a mistake.
- Know the totals: coins give outcomes, two dice give , a pack gives .
- Two coins give — four outcomes. Exactly one head is , at least one head , no head , two heads .
- Three coins give outcomes: exactly two heads is , at least two heads .
- One die: even , prime , multiple of is , greater than is . *At most * is but *less than * is .
- Two dice: sum is , sum is , a doublet , sum is , sum above is , sum is .
- Order matters: and are different outcomes, which is why and not .
- Cards: a king , a heart , a red card , a face card , the ace of spades , a red king .
- Balls: red, blue, green give , and , which total . And white with black give and .
- Complement rule: . So gives ; ; .
- **Use the complement whenever you see at least or not**, and name it in words first — the complement of red is blue or green.
- Finding a count: red with gives , so .
- A set of non-overlapping events covering everything must total **exactly .
- A fair coin has no memory** — five tails do not make a head more likely next time.
Write out all outcomes for two dice once, by hand, and count how many give each sum from to — the shape that appears is worth more than any formula you could memorise instead.