2. A straight is a hand in which the cards are in consecutive order by rank. For

Question

2. A straight is a hand in which the cards are in consecutive order by rank. For the purposes of the following question, going around the end of the rank order is not allowed. (For example, 4 3 2 A K would not count as a straight.)

a. Suppose you draw a five-card hand randomly from the deck and get four cards that that would make a straight if you could replace the fifth card. (e.g. J   10 9 8 3 or K 7 6 4 3). If you are allowed to discard the fifth card and draw one at random from the remaining 47 cards in the deck, what is the probability that your modified hand will be a straight? [3]

      Hint: There are several cases to consider . . .

       A flush is a hand in which all the cards are from the same suit.

b.Suppose you draw a five-card hand randomly from the deck. What is the probability that this hand is a flush? [1]

c. Suppose you draw a five-card hand randomly from the deck. What is the probability that this hand is both a straight and a flush? [1]

Explanation / Answer

The four consecutive pairs could be:

( A 2 3 4) ( 2 3 4 5 ) ( 3 4 5 6 ) (4 5 6 7) ( 5 6 7 8) (6 7 8 9) ( 7 8 9 10) ( 8 9 10 J ) ( 9 10 J Q) (10 J Q K) (J Q K A )

( Q K A 2) ( K A 2 3).

Now, for the following pairs, either a card before or after would make a straight

( 2 3 4 5 ) ( 3 4 5 6 ) (4 5 6 7) ( 5 6 7 8) (6 7 8 9) ( 7 8 9 10) ( 8 9 10 J ) ( 9 10 J Q) (10 J Q K) which means there are 8 possible cards for each of the above pairings.

Hence, probability = (9/13) * (8/47)

For, (A 2 3 4) and (J Q K A ) only one card is suitable. Hence 4 possible cards for each pair

Probability : (2/13) * (4/47)

For the other two pairings, we don't have a card that can make a straight.

So, Probability : (2/13) * (0/47)

Total : (9/13) * (8/47) + (2/13) * (4/47) + (2/13) * (0/47)

= 80 / 611

.

B)

Probability of flush = 13C5 * 4C1 / 52C5

C)

Probability of Straight flush = 10C1 * 4C1 / 52C5

Hope this helps.

Refer to this page for more explanation:

https://en.wikipedia.org/wiki/Poker_probability

Related Questions

Navigate

• Browse (All)
• Browse 2
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.