Question

A and B play a game where each is asked to select a number from $1$ to $25$. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trail is
$A)\dfrac{1}{{25}}$
$B)\dfrac{{24}}{{25}}$
$C)\dfrac{2}{{25}}$
$D)$ None of these

Answer 1

Let us first find the total number of ways in which numbers can be selected by A and B, then we will find the number of ways in which either player can choose the same answer.
By the use of the probability formula, we will find the game-winning in a single trial, then subtract the probability to win the game in a single trial from the total probability which is $1$ to get the probability that the game is not won in a single trial.
Formula used:

$P = \dfrac{F}{T}$where P is the overall probability, F is the possible favorable events and T is the total outcomes from the given.

Complete step-by-step solution:
From the given question, we are supposed to find the total number of ways in which numbers can be chosen from A and B. the total number is $25$ because we were asked to select a number from $1$ to $25$.
Both of them have access to choose all the numbers without any restriction.
Hence the total number of ways in which numbers can be chosen by A and B is $25 \times 25 = 625$ ways.
Now we will find the number of ways in which either player can choose the same numbers, which is $25$ (because there is no restriction in choosing the numbers, so maybe both of them will choose the same set of numbers)
Thus, to find the probability of winning the prize which is number chosen is same is given by the ratio of the favorable events is $P = \dfrac{F}{T} \Rightarrow \dfrac{{25}}{{625}} = \dfrac{1}{{25}}$
Now, to get the probability that A and B will not choose the same number and don’t win the prize is given by the subtraction of the probability to win the game in a single trial with the number $1$.
Hence, we get $1 - \dfrac{1}{{25}} = \dfrac{{25 - 1}}{{25}} = \dfrac{{24}}{{25}}$ which is the probability that they will not win a prize in a single trial.
Therefore, the option $B)\dfrac{{24}}{{25}}$ is correct.

Note: First, let us assume the overall total probability value is $1$ (this is the most popular concept that used in the probability that the total fraction will not exceed $1$ and everything will be calculated under the number $0 - 1$ as zero is the least possible outcome and one is the highest outcome)
Thus, by using this concept, let us take the winning and losing probability and add them we get $\dfrac{1}{{25}} + \dfrac{{24}}{{25}} = \dfrac{{25}}{{25}} = 1$ , and hence the total probability will be not exceeding $1$ is true.
$\dfrac{{24}}{{25}}$which means the favorable event is $24$ and the total outcome is $25$