Question

Draw one card with replacement from a well-shuffled standard poker deck until you draw a spade. Find the probability of each of the following:

a. You draw only once.

b. You draw at least once.

c. You draw at least twice.

d. You draw at least three times, but no more than five times.

Answer 1

The probability of having the first success in trial is modelled by geometric distribution given by

where x = 1,2,3.....

Here p is the probability of success.

There are 13 spade cards in 52 cards.

Therefore the probability of picking a spade = 13/52 = 0.25

a) The probability that you pick only once to get spade = P(X = 1) = p = 0.25

b) Probability that you pick atleast once to get spade = P(X = 1) + P(X = 2) + P(X = 3) + ........ = 1

Since here you have included entire sample space

c) Probability that you pick atleast twice to get spade = P(X = 2) + P(X = 3) + P(X = 4) + ........ = 1 - P(X = 1) = 1 - 0.25 = 0.75

d) Probability that you pick atleast 3 times but no more than 5 times to get spade = P(X = 3) + P(X = 4) + P(X = 5) = = = 0.3252

Therefore the probability that you pick atleast 3 times but no more than 5 times to get spade is 0.3252