SOLUTION: Compute the probability that a 5-card poker hand
a. is dealt to you that contains all hearts.
b. hand is dealt to you that contains four Aces.
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-> SOLUTION: Compute the probability that a 5-card poker hand
a. is dealt to you that contains all hearts.
b. hand is dealt to you that contains four Aces.
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Question 1201659: Compute the probability that a 5-card poker hand
a. is dealt to you that contains all hearts.
b. hand is dealt to you that contains four Aces.
13 hearts out of 52 cards total
13/52 = probability of getting a heart
12/51 = probability of getting a second heart
We do not put the cards back, aka no replacement.
Subtract 1 from numerator and denominator each time we take out another card.
11/50 = probability of getting a third heart
10/49 = probability of getting a fourth heart
9/48 = probability of getting a fifth heart
Multiply out the fractions
(13/52)*(12/51)*(11/50)*(10/49)*(9/48)
= (13*12*11*10*9)/(52*51*50*49*48)
= 154440/311875200
= 33/66640 is the probability of getting 5 hearts in a row.
Another approach:
n = 13 hearts
r = 5 slots to fill
Use the nCr combination formula. The order doesn't matter in a poker hand.
n C r = (n!)/(r!(n-r)!)
13 C 5 = (13!)/(5!*(13-5)!)
13 C 5 = (13!)/(5!*8!)
13 C 5 = (13*12*11*10*9*8!)/(5!*8!)
13 C 5 = (13*12*11*10*9)/(5!)
13 C 5 = (13*12*11*10*9)/(5*4*3*2*1)
13 C 5 = 154440/120
13 C 5 = 1287
There are 1287 ways to get a hand of nothing but hearts.
Follow similar steps to find 52C5 = 2598960 different five-card hands.
1287/2598960 = 33/66640
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Part (b)
There is 1 way to select the 4 aces where order doesn't matter.
There are 52-4 = 48 other cards that aren't an ace.
Ultimately there are 1*48 = 48 five-card hands that consist of 4 aces and some other card.
This is out of the 2598960 different five-card hands.
48/2598960 = 1/54145 represents the probability of getting four aces.
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