1.11. Problem. (Sections 2.2-2.4, 3.1) Five cards are drawn from a standard deck of 52 cards....

1.11. Problem. (Sections 2.2-2.4, 3.1) Five cards are drawn from a standard deck of 52 cards.

(a) Given that exactly three of the five cards show a hearts suit, calculate the probability that the hand also includes a three-of-a-kind.

(b) Given that the five card hand contains a three-of-kind, find the probability that it contains at three hearts.

Solutions

Expert Solution

As per my knowledge of cards a three-of-a-kind means cards of same denominations.

P[ 5 out of 3 cards are heart ] = 13C3*39C2 ( 3 heart cards and 2 any other cards )

P[ a three-of-a-kind ] = 13*4C3*48C2 ( 3 cards from any 4 cards of same denomination and 2 cards from other 48 cards , also multiplies by 13 because there are 13 options for the card to be chosen )

P[ 5 out of 3 cards are heart and a three-of-a-kind ] = 3*13C3*3C2 ( any 3 heart and out of those 3 only any one card is repeated twice )

a) Given that exactly three of the five cards show a hearts suit, calculate the probability that the hand also includes a three-of-a-kind.

P[ a three-of-a-kind | 5 out of 3 cards are heart ] = P[ 5 out of 3 cards are heart and a three-of-a-kind ] / P[ 5 out of 3 cards are heart ] = 3*13C3*3C2 / 13C3*39C2 = 3*3C2 / 39C2 = 9/741 = 0.0121

P[ a three-of-a-kind | 5 out of 3 cards are heart ] = 0.0121

b) (b) Given that the five card hand contains a three-of-kind, find the probability that it contains at three hearts

P[ 5 out of 3 cards are heart | a three-of-a-kind ] = P[ 5 out of 3 cards are heart and a three-of-a-kind ] / P[ a three-of-a-kind ] = 3*13C3*3C2 / 13*4C3*48C2 = 3*3*286 / 13*4*1128 = 2574 / 58656 = 0.0439

P[ 5 out of 3 cards are heart | a three-of-a-kind ] = 0.0439

milcah answered 8 months ago

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