Introductory Statistics 10th ed. (Weiss, 2016) p.222 (Chapter 4): Case Study Dis

Question

Introductory Statistics 10th ed. (Weiss, 2016) p.222 (Chapter 4):

Case Study Discussion Texas Hold'em

At the beginning of the chapter on pages 156-157, we discussed Texas Hold'em and described the basic rules of the game. Here we examine some of the simplest probabilities associated with the game.

Recall that, to begin, each player is dealt 2 cards face down, called 'hole cards," from an ordinary deck of 52 playing cards, as pictured in Fig. 4.3 on page 165. The best possible starting hand is two aces, referred to as "pocket aces."

a. The probability that you are dealt pocket aces is 1/221, or 0.00452 to three significant digits. If you studied either Sections 4.5 (Conditional Probability) and 4.6 (The Multiplication Rule; Independence) or Section 4.8 (Counting Rules), verify that probability.

b. Using the result from part (a), obtain the probability that you are dealt "pocket kings."

c. Using the result from part (a) and your analysis from part (b), find the probability that you are dealt a "pocket pair." that is, two cards of the same denomination.

Next recall that, after receiving your hole cards, there is a betting round. Subsequently, 3 cards, called "the flop," are dealt face up in the center of the table. To do the remaining problems, you need to have studied either Sections 4.5 (Conditional Probability) and 4.6 (The Multiplication Rule; Independence) or Section 4.8 (Counting Rules). Assuming that you are dealt a pocket pair, determine the probability that the flop

d. contains at least 1 card of your denomination. (Hint: Complementation Rule.)

e. gives you "trips," that is, contains exactly 1 card of your denomination and 2 other unpaired cards.

f. gives you "quads," that is, contains 2 cards of your denomination.

g. gives you a "boat," that is, contains 1 card of your denomination and 2 cards of another denomination.

I need these answers asap. Much mahalo!!

Explanation / Answer

a)

There are 52C2 = 1326 ways to get any 2 cards.
There are 4C2 = 6 ways to get 2 aces.

Hence,

P(2 aces) = 6/1326 = 1/221 [ANSWER]

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b)

There are also 4C2 = 6 ways to get 2 kings. So,

P(2 kings) = 1/221 [ANSWER, still the same]

*****************

C)

There are 13 denominations possible, so

P(pocket pair) = 13*(1/221) = 1/17 [ANSWER]

******************

d)

There are 50 cards left to chose the flop from.

Note that

P(at least one) = 1 - P(none)

Thus, there are 48 cards not in your denomination. There are 48C3 = 17296 ways to choose them. There are 50C3 = 19600 ways to choose the flop.

Hence,

P(at least one) = 1 - 17296/19600 = 0.11755102 [ANSWER]

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