Question

Step 1: Doyle, a professional poker player, is at the poker table. As he sits at the table looking at his hand and at the upturned cards on the table, Doyle has seen 24 cards, and the only hearts are the 2 in his hand. None of other players have any hearts. The full deck contains 13 hearts among its 52 cards. Doyle is hoping the next three cards dealt to him will be hearts. This outcome would give him 5 cards from the same suit, a hand that is called a flush. Given below is a tree diagram describing the outcomes for the two cards that he will be dealt, on which three of the probabilities (a, b and c) are designated. At the first stage, his draw can be a heart or a non-heart. At the second stage, he has the same possible outcomes but the probabilities are different. First card Second card Third card Heart Heart Heart < Not Not Heart Not Heart Heart Not Not b Not Heart Not What is the probability of outcome a? Give your answer to 3 decimal places.

Answer 1

Number of cards in deck = 52

Number of cards Doyle has seen = 24

Number of cards Doyle has not seen = 52-24=28

Number of heart cards in deck = 13

Number of heart cards with Doyle = 2

Number of heart cards left in deck = 13-2=11

Step 1 :

a :Probability of second card is not hearts given that first card is not hearts

Number of cards left in the deck after the first card is dealt = 28-1 = 27

Since the first dealt is not hearts, number of heart cards in the deck = 11

i.e Number of non-hearts cards in the deck = 27- 11 =16

a :Probability of second card is not hearts given that first card is not hearts

= Number of non-hearts cards in the deck / Number of cards left in the deck after the first card is dealt = 16/27 = 0.593

a = 0.593

Step 2 :

b : Probability of third card is hearts given that first card is not hearts and second card is not hearts

Number of cards left in the deck after the first and second cards are dealt = 28-1-1 = 26

Since the first dealt is not hearts and second card is not hearts, number of heart cards in the deck = 11

Probability of third card is hearts given that first card is not hearts and second card is not hearts

= Number of hearts cards in the deck / Number of cards left in the deck after the first and second card are dealt = 11/26 = 0.423

b=0.423

Step 3 :

c : Probability of third card is not hearts given that first card is not hearts and second card is not hearts

This is complement of b

c = 1-b = 1-0.423 = 0.577;

Number of cards left in the deck after the first and second cards are dealt = 28-1-1 = 26

Since the first dealt is not hearts and second card is not hearts, number of heart cards in the deck = 11

Number of non-hearts cards in the deck = 26- 11 =15

Probability of third card is not hearts given that first card is not hearts and second card is not hearts

= Number of non-hearts cards in the deck / Number of cards left in the deck after the first and second cards are dealt = 15/26 = 0.577

c=0.577