Five cards are dealt from a shuffled deck of 52. Find the probability that:
a) The hand has 3 spades and 2 hearts.
b) The hand has 2 aces, 2 kings, and one other card.
a) prob = C(13,3) * C(13,2)/C(52,5)
= 286*78/2598960
= 286/33320
= 11/99960
b) prob = C(4,2) * C(4,2)* C(44,1)/C(52,5)
= 6*6*44/2598960
= 3*11/54145
= 33/54145
is C for nCr???? i'm a little confused
yes
C(n,r) = nCr
its an easier notation to type
Ohhhhhhh! okay, thank you! :)
Wait, how'd you get 33320 in A?
2598960/78 = 33320
To find the probability of different card combinations in a hand, we first need to determine the total number of possible hands. Then we calculate the number of favorable outcomes (i.e., the hands that meet the given conditions). Finally, we divide the number of favorable outcomes by the total number of possible hands to obtain the probability.
a) The hand has 3 spades and 2 hearts.
First, we need to determine the total number of possible hands. There are 52 cards in a deck, and we are selecting 5 cards without replacement. Therefore, the total number of possible hands is given by the combination formula: C(52, 5) = 52! / (5!(52-5)!) = 2,598,960.
Next, let's calculate the number of favorable outcomes, which are the hands with 3 spades and 2 hearts.
The number of ways to choose 3 spades from the 13 spades in the deck is given by: C(13, 3) = 13! / (3!(13-3)!) = 286.
Similarly, the number of ways to choose 2 hearts from the 13 hearts in the deck is given by: C(13, 2) = 13! / (2!(13-2)!) = 78.
Since the events of choosing spades and hearts are independent, we multiply these two numbers together to get the total number of favorable outcomes: 286 * 78 = 22,308.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible hands: 22,308 / 2,598,960 ≈ 0.0086.
Therefore, the probability of getting a hand with 3 spades and 2 hearts is approximately 0.0086.
b) The hand has 2 aces, 2 kings, and one other card.
As before, we first determine the total number of possible hands, which is the same as above: 2,598,960.
Next, we calculate the number of favorable outcomes, which is the number of ways to choose 2 aces from the 4 aces in the deck (C(4, 2) = 6), multiplied by the number of ways to choose 2 kings from the 4 kings in the deck (C(4, 2) = 6), and finally multiplied by the number of ways to choose one other card from the remaining 44 cards in the deck (C(44, 1) = 44).
Hence, the total number of favorable outcomes is: 6 * 6 * 44 = 1,584.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible hands: 1,584 / 2,598,960 ≈ 0.00061.
Therefore, the probability of getting a hand with 2 aces, 2 kings, and one other card is approximately 0.00061.