Question

Find the probability that a number selected from the number 1 to 25 is not a prime number, when each of the given numbers are equally likely to be selected.

Answer 1

Hint: For solving this question, we need to know about the basic concepts of probability. To find the probability for this problem, we first find the total number of outcomes possible. Then we find the desired number of outcomes for the particular question and then use the definition of probability to find the answer.

Complete step-by-step answer:
First, we start with the meaning of the term probability. Probability helps us to find out the likelihood of an event to occur. For example, if in a weather forecast, it is said that there is 80% chance for the day to be rainy, this means that the probability for it to rain on that particular day is 0.8 (basically, we convert percentage to decimal by dividing by 100). Another example is that of a coin toss. The total number of outcomes in this case is 2 (heads and tails). To get the desired outcome (say heads), the probability is 0.5 since it is equally likely to be heads or tails.
Now, coming onto the above question. First, we find out the total number of outcomes. The number of outcomes would be 25 (that is, counting from 1 till 25). Now, we count the desired number of outcomes. Since, the number should not be a prime number (prime number means that the number should have no other divisor except 1 and the number itself). Thus, the desired outcomes are – 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25 (thus, there are 16 outcomes).
Probability = $\dfrac{\text{Desired number of outcomes}}{\text{Total number of outcomes}}$
Probability = $\dfrac{16}{25}$
Hence, the required probability is $\dfrac{16}{25}$.

Note: An alternative to solve the problem is to use the following equation below-
Probability (number is a prime) + Probability (number is not a prime) = 1 -- (1)
Now, we need to find the probability of the number not to be a prime number. Thus, we can calculate the probability of the number to be a prime and then subtract that probability from 1. In this case, also the number of outcomes is 25. The desired number of outcomes is 9 (that is 2,3,5,7,11,13,17,19,23). Thus, calculating the probability for number to be a prime is-
Probability = $\dfrac{\text{Desired number of outcomes}}{\text{Total number of outcomes}}$
Probability = $\dfrac{9}{25}$
Now, putting in expression (1), we get,
$\dfrac{9}{25}$ + Probability (number is not a prime) = 1
Probability (number is not a prime) = $\dfrac{16}{25}$
Hence, we can get back the same required probability by this method also.