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