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Question: In a lottery of 50 tickets numbered from 1 to 50, one ticket is drawn. Find the probability that the...

In a lottery of 50 tickets numbered from 1 to 50, one ticket is drawn. Find the probability that the drawn ticket bears a prime number.

Explanation

Solution

To solve this question first we need to list all the favorable outcomes by calculating the prime numbers from 11 to 5050. Then we count the total number of outcomes as lottery tickets are numbered from 1 to 50 and use the formula to calculate the probability. Following formula is used-
P(A)=n(E)n(S)P\left( A \right)=\dfrac{n\left( E \right)}{n\left( S \right)}
Where, A is an event,
n(E)=n\left( E \right)= Number of favorable outcomes and n(S)=n\left( S \right)= number of total possible outcomes

Complete step-by-step solution:
We have been given that in a lottery of 50 tickets numbered from 1 to 50, one ticket is drawn.
We have to find the probability that the drawn ticket bears a prime number.
Now, we know that the probability of an event is given by the formula
P(A)=n(E)n(S)P\left( A \right)=\dfrac{n\left( E \right)}{n\left( S \right)}
Where, A is an event,
n(E)=n\left( E \right)= Number of favorable outcomes and n(S)=n\left( S \right)= number of total possible outcomes
Now, we have 50 tickets numbered from 1 to 50, so total number of possible outcomes will be n(S)=50n\left( S \right)=50
Now, we have 1515 prime numbers from 1 to 50.
So, the number of favorable outcomes will be n(E)=15n\left( E \right)=15
Now, the probability that the one drawn ticket bears a prime number will be
P(A)=n(E)n(S)P\left( A \right)=\dfrac{n\left( E \right)}{n\left( S \right)}
P(A)=1550 P(A)=310 \begin{aligned} & P\left( A \right)=\dfrac{15}{50} \\\ & P\left( A \right)=\dfrac{3}{10} \\\ \end{aligned}
So, the probability that the one drawn ticket bears a prime number is 310\dfrac{3}{10}.

Note: A prime number has only two distinct factors, one and the number itself. 22 is the only prime number which is even, all other prime numbers are odd numbers. The number of 11 is not a prime number. Some students may count 11 also in the list of prime numbers and get an incorrect answer. The list of prime numbers from 1 to 50 is 2,3,5,7,11,13,17,19,23,29,31,37,41,43,472,3,5,7,11,13,17,19,23,29,31,37,41,43,47.