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Question

A pair of dice is thrown 4 times. If getting a doublet is considered as a success,
(1) find the probability of getting a doublet.
(2) hence, find the probability of two successes.

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Solution

(a)
Possible outcomes is S={(1,1),(1,2)....(1,6),(2,1),....(2,6),(3,1),....(3,6),.......(6,6)}
Total no. of outcomes is n(S)=36
Outcomes with doublets is A={(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
Then probability of getting a doublet is P(A)=n(A)n(S)=636=16=a
(b)
Let B an event of not getting successes
Then P(B)=1P(A)=116=56=b
Let X be the number of success.
If we get two successes, then the no. of failures is 2 as a pair of dice is thrown 4 times.
Then P(X=2,2 successes, 2 failures)=4C2 a2 b2
=6×136×2536=25216

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