Question

a die is thrown thrice.Getting a 5 or 6 is considered as success.Find the probability of getting

(1-)exactly 2 successes

(2-)at most 2 successes

(3-)atleast 2 successes

Answer 1

1) For exactly 2 successes, on unsuccess may be obtained at any of the three places. Also the unsuccess can be any of the 4 numbers i.e. 1, 2, 3, 4. Also 2 successes can occur in 4 ways viz. {55, 66, 56, 65}.So if E represents the events here then,

| E | = 3 x 4 x 4 = 48

Also die is thrown 3 times, so | S | = 6⁸ = 216

So, Probability of getting exactly 2 successes P(E) = | E | / | S | = 48 / 216 = 2/9 Ans.


3) Getting no success is same thing as getting 3 successes which can be obtained in 2 x 2 x2 = 8 ways,
as the success here is obtained in 2 ways (as a 5 or as a 6).

So here probability of getting no success = 8/216 = 1/27

P(at least 2 success) = P(2 success or 3 success) = P(2 success ) + P(3 success)

Hence the event of getting 2 successes is disjoint from the event of getting 3 successes.
Probability of getting at least 2 successes = 2/9 + 1/27 = 7/27 Ans.


2) We see that one success can be obtained in P(3, 1) x P(3, 2) = 3 x 3 = 9 ways
So the probability of getting 2 most successes = (9 + 7) / 27 = 16/27 Ans.