Q. 51

A die is thrown 5 times. Find the probability that an odd number will come up exactly three times.

Answer :

Given that, a die is thrown 5 times.


Let p be the probability of getting an odd number in a throw.


Since there are 6 possible outcomes in a throw of a die. That is, {1, 2, 3, 4, 5, 6}


And there are 3 odd numbers out of these 6 outcomes. That is, {1, 3, 5}




Then, let q be the probability of getting an even number in a throw.


And we know that, p + q = 1


q = 1 – p




Let X be a random variable representing a number of odd numbers in n throw of a die.


Then, the probability of getting r odd numbers out of n throw of the die can be given as,


P (X = r) = nCrprqn-r


Here, n = 5


Putting values of n, p, and q in the above formula. We get




…(i)


We need to find the probability that an odd number will come up exactly three times.


It is given by,


Probability = P (X = 3)


Put r = 3 in equation (i) to find P (X = 3), we get








Thus, the probability of getting an odd number to come up exactly three times is 5/16.


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