Q. 484.0( 3 Votes )

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes.

Answer :

Given that, a pair of dice is thrown 4 times.


And a doublet is considered as a success.


Let p be the probability of getting a doublet in a throw of a pair of dice.


Since, there are 36 possible outcomes in total. {(1, 1), (1, 2), (1, 3), …, (1, 6), ..., (2, 6), …, (6 ,6)}


And the 6 possible doublets in 36 outcomes. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}


So,




And also, p + q = 1


q = 1 – p





Let X denote a random variable representing a number of doublets (successes) out of 4 throws.


So, Binomial distribution of getting r successes out of 4 throws is given by


P (X = r) = nCrprqn-r


Here, n = 4.


Now, substituting values of n, p and q in the formula P (X = r). We get


…(i)


We need to find the probability distribution of the number of successes.


The probability of 0 success in 4 throws is given by,


Probability = P (X = 0)


Put r = 0 in (i),






The probability of 1 success in 4 throws is given by,


Probability = P (X = 1)


Put r = 1 in (i),







The probability of 2 successes in 4 throws is given by,


Probability = P (X = 2)


Put r = 2 in (i),







The probability of 3 successes in 4 throws is given by,


Probability = P (X = 3)


Put r = 3 in (i),






The probability of 4 successes in 4 throws is given by,


Probability = P (X = 4)


Put r = 4 in (i),





Thus, the probability distribution is



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