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# Two digits numbers are formed using the digits 0, 1, 2, 3, 4 where digits are not repeated. Find the probability of the events that the number so formed is a prime number

Last updated date: 23rd Jul 2024
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Hint: Here the given question is based on the concept of probability. We have to find the probability of choosing an event that the two digit prime number formed in a given set of numbers, to first find the set of possible two digit numbers in a given set of numbers, by using the definition of probability and on further simplification we get the required probability.

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are to happen, using it. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.
The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.
$Probability of event to happen P\left( E \right) = \dfrac{{Number of favourable outcomes}}{{Total Number of outcomes}}$
Consider the given set of numbers: 0, 1, 2, 3, 4.
Now, forming a possible two digits number in the given set of number i.e.,
$\therefore \,\,\,S = \left\{ {10,12,13,14,20,21,23,24,30,31,32,34,40,41,42,43} \right\}$
Total number of two digits number are = 16
Now, the A is the event that the number so formed is prime number i.e.,
$A = \left\{ {13,23,31,41,43} \right\}$
Total number of prime numbers = 5
By, the definition of probability
$\Rightarrow \,\,P\left( {prime numbers} \right) = \dfrac{{Number of prime numbers}}{{Total Number of two digits numbers}}$
$\Rightarrow \,\,P\left( {prime numbers} \right) = \dfrac{5}{{16}}$
Hence, the probability of the event that the number formed is a $\dfrac{5}{{16}}$.
So, the correct answer is “$\dfrac{5}{{16}}$”.

Note: The probability is a number of possible values. Candidates must know we have to use the permutation concept or combination concept to solve the given problem because it is the first and main thing to solve the problem. Here we are arranging the things in the possible ways so we are using the concept of permutation.