Question

A card is drawn 5 times in a row (replaced every time). What is the probability that there will be

a) at least 3 pictures?

b) at least 3 pictures or all 5 black cards?

Answer 1

Binomial distribution: P(X) = nCx p^x q^n-x

a) P(picture card), p = (3x4)/52

= 3/13

q = 1 - p = 10/13

Number of trials, n = 5

P(at least 3 pictures) = P(3) + P(4) + P(5)

= 5C3x(3/13)³x(10/13)² + 5x(3/13)⁴x(10/13) + (3/13)⁵

= 0.0727 + 0.0109 + 0.0007

= 0.0843

b) P(A or B) = P(A) + P(B) - P(A & B)

P(at least 3 pictures or all 5 black cards) = P(at least 3 pictures) + P(all 5 black cards) - P(at least 3 black pictures)

= 0.0843 + (1/2)⁵ - [P(3 black pictures) + P(4 black pictures) + P(5 black pictures)]

= 0.0843 + 0.0313 - [5C3x(3/26)³x(23/26)² + 5x(3/26)⁴x(23/26) + (3/26)⁵]

= 0.0843 + 0.0313 - 0.0128

= 0.1028