Question

A hand consists of 4 cards from a well-shuffled deck of 52 cardsa. Find the total number of possible 4 card poker handsb. A heart flush is a 4 card hand consisting of all heart cards. Find the number of possible heart flushes.c. Find the probability of being dealt a heart flusha. There are a total ofpoker hands

Answer 1

`\begin{gathered} a\mathrm{})\text{ Find the total number of possible 4 card poker hands} \\ \text{There are 52 cards, so that means we have }n=52\text{ to choose from and we get} \\ r=4\text{ cards to choose} \\ 52C4=\binom{52}{4}=(n!)/((r!)(n-r)!) \\ \binom{52}{4}=(52!)/((4!)(52-4)!)=(52!)/((4!)(48!)),\text{ remember the }48!\text{ cancels the }48!\text{ in the numerator} \\ \binom{52}{4}=(52\cdot51\cdot50\cdot49)/(4\cdot3\cdot2\cdot1) \\ \binom{52}{4}=(6497400)/(24)=270725 \\ \text{There are 270725 number of possible 4 card poker hands.} \\ \\ b.)\text{ Find the number of possible heart flushes.} \\ \text{There are 13 hearts in a suit, so that means we have }n=13\text{ to choose from and we get} \\ r=4\text{ cards to choose} \\ 13C4=\binom{13}{4}=(13!)/((4!)(13-4)!) \\ \binom{13}{4}=(13!)/(4!\cdot9!)=(13\cdot12\cdot11\cdot10)/(4\cdot3\cdot2\cdot1)=(17160)/(24) \\ \binom{13}{4}=715 \\ \text{There are 715 possible heart flushes.} \\ \\ c)\text{Find the probability of being dealt a heart flush} \\ \text{Divide the result from in b with a} \\ (715)/(270725)=0.002641056423 \\ \text{Rounded off to 5 decimal places, the probability of being dealt with a heart flush is} \\ 0.00264 \end{gathered}`