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Question

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4
(ii) six appears on at least one die

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Solution

When a die is tossed two times, we obtain (6×6)=36 number of observations.
Let X be the random variable, which represents the number of successes.
(i) Here, success refers to the number greater than 4.
P(X=0)=P(number less than or equal to 4 on both the tosses)=46×46=49
P(X=1)=P(number less than or equal to 4 on first toss and greater than 4 on second toss)+
P(number greater than 4 on first toss and less than or equal to 4 on second toss)
=46×26+46×26=49
P(X=2)=P(number greater than 4 on both the tosses)=26×26=19
Thus, the probability distribution is as follows.
(ii) Here, success means six appears on at least one die.
P(Y=0)=P(six does not appear on any of the dice)=56×56=2536
P(Y=1)=P(six appears on at least one of the dice)=1136
Thus, the required probability distribution is as follows.

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