Question

Three cards are randomly selected from a fair deck (without replacement). Find the probabilities of the following events by applying the definition of conditional probability: • All hearts, given all cards are red.

• All hearts, given one of the cards is a king

• All hearts, given no spades.

• All hears, given two kings.

Answer 1

Bayes' theorem: P(A | B) = P(A and B)/P(B)

Number of ways to choose r items from n given items iss given by nCr = n!/(r! X (n-r)!)

Number of red cards = 26

Number of hearts = 13

P(all hearts | all red) = 13C3 / 26C3

= 0.11

P(all hearts | one card is a kind) = P(all hearts and one king) / P(one king)

= (3 x 1/52 x 12/51 x 11/50) / (1 - P(no kings))

= 0.003/(1 - 48C3/52C3)

= 0.003/(1 - 0.783)

= 0.014

Number of non spade cards = 52-13 = 39

P(all hearts | no spades) = 13C3/39C3

= 0.031

P(all hearts | two kings) = 0, because there is only one king of hearts