Question

A die is thrown once. Find the probability of getting a number between 2 and 6.

Answer 1

Hint: Use the fact that if E is the event and S the sample space of the random experiment, then the probability of the event E is given by $P\left( E \right)=\dfrac{number\ of\ favourable\ cases}{Total\ number\ of\ cases}=\dfrac{n\left( E \right)}{n\left( S \right)}$. Consider the random experiment of throwing a die and assume that E is the event of getting a number between 2 and 6. We know that a die has six numbers from 1 to 6. Using this, find n(E) and n(S) and hence find the probability of event E.

Complete step-by-step answer:
Let X be the random experiment of tossing a die. Let S be the sample space of the random experiment and E the event of getting a number between 2 and 6.
Hence, we have
$S=\left\{ 1,2,3,4,5,6 \right\}$ and $E=\left\{ 3,4,5 \right\}$
Hence, we have n(S) = 6 and n(E) = 3
Hence, by the definition of probability, we have
$P\left( E \right)=\dfrac{3}{6}$
Simplifying, we get
$P\left( E \right)=\dfrac{1}{2}$
Hence the probability of getting a number between 2 and 6 when a fair die is cast is 0.5

Note: [1] The die has to be fair(i.e. equal chance of occurrence of all the numbers) for the application of the above formula. If the die is biased, then the above formula cannot be used. In that case, we used the conditional probability of an event.
[2] There is a mistake of overcounting done by many students as they include both 2 and 6 in the event E. This is incorrect since we are asked to find the numbers between 2 and 6, i.e. greater than 2 and less than 6. Unless mentioned (both inclusive, left inclusive, etc.), we should not add the boundary numbers to our set.