No Probability Can Ever Be Bigger Than One

Never. A probability is the fraction of outcomes that are favourable, so at most all of them can be — and that gives exactly 1. An answer of $\frac{7}{6}$ or $1.5$ means a mistake has been made, not an unusually likely event.

That single check catches most errors in the chapter. This page covers everything in the ICSE Class 7 Mathematics chapter on probability: chance and randomness, experimental probability, theoretical probability, and the complement of an event.

An event is certain if it must happen, impossible if it cannot happen, and likely or unlikely if it may happen.

- Certain — the sun will set today; a tossed coin will show heads or tails. Probability 1.
- Impossible — rolling a 7 on an ordinary die; drawing a green ball from a bag of only red balls. Probability 0.
- Likely — rain during the monsoon in Mumbai.
- Unlikely — rain in Jaisalmer in May.
- Equally likely — a tossed coin showing heads rather than tails.

An experiment is random when its outcome cannot be predicted in advance even though all the possible outcomes are known. Rolling a die is random; the six possibilities are known, but which one appears is not.

So probability measures how likely, on a scale from 0 to 1, where the two ends are the special cases of impossible and certain.

The word equally likely carries real weight and is often assumed carelessly. A die is equally likely to show any face only if it is fair — a weighted die has the same six outcomes without them being equally likely, and then the counting method in the later section would not apply.

Experimental (empirical) probability is worked out from what actually happened in a set of trials:

$$P(\text{event})=\frac{\text{number of trials in which the event occurred}}{\text{total number of trials}}$$

Worked example. A coin is tossed 100 times and heads appears 54 times:

$$P(\text{head})=\frac{54}{100}=0.54$$

and tails appeared $100-54=46$ times, so $P(\text{tail})=\frac{46}{100}=0.46$. The two add to 1, as they must.

Worked example with a die. A die is thrown 60 times and a 6 comes up 8 times:

$$P(6)=\frac{8}{60}=\frac{2}{15}$$

Worked example from a survey. Of 40 students asked, 15 chose cricket:

$$P(\text{cricket})=\frac{15}{40}=\frac{3}{8}$$

The important feature is that experimental probability depends on the trials. Toss the same coin another 100 times and you may get 48 heads instead of 54, giving a different value.

That is exactly how it differs from the theoretical figure. The theoretical probability of a head is $\frac{1}{2}$, and the experimental result of 0.54 is close to it but not equal — and the more trials you perform, the closer the experimental value tends to sit to the theoretical one.

The complement of an event $E$, written $E^{\prime }$, is the event that $E$ does not happen. Together they cover every possibility, so

$$P(E)+P(E^{\prime })=1\text{and therefore}P(E^{\prime })=1-P(E)$$

Worked example. From the bag of 5 red, 3 blue and 2 green marbles, $P(\text{red})=\frac{1}{2}$, so

$$P(\text{not red})=1-\frac{1}{2}=\frac{1}{2}$$

Check by counting directly: the non-red marbles are the 3 blue and 2 green, giving $\frac{5}{10}=\frac{1}{2}$. The two methods agree.

Worked example with a die. $P(6)=\frac{1}{6}$, so

$$P(\text{not }6)=1-\frac{1}{6}=\frac{5}{6}$$

The complement rule is often the quicker route. To find the probability of not getting a 1 on a die, subtracting $\frac{1}{6}$ from 1 beats listing the five other faces.

From this rule the whole range follows. Since $P(E)$ and $P(E^{\prime })$ are both counts of outcomes they cannot be negative, and since they add to 1 neither can exceed 1. So

$$0\le P(E)\le 1$$

with 0 for an impossible event and 1 for a certain one — which is why the answer at the top of this page is never.