An ordinary deck of playing cards has 4 suits, with 13 cards of each suit. In many games, each of 4 players is dealt 13 cards at random.
a. What is the probability that such a hand has each of the following?
1. Exactly 5 spades
2. Which is more probable, getting all 4 aces or getting all 13 cards of the same suit? Justify your answer.
a. To find the probability of getting exactly 5 spades in a hand of 13 cards, we need to calculate the ratio of favorable outcomes to total possible outcomes.
Total possible outcomes = Number of ways to select 13 cards from a deck of 52 cards
Favorable outcomes = Number of ways to select 5 spades from the 13 spades available, multiplied by the number of ways to select 8 non-spade cards from the remaining 39 cards.
1. Calculation:
Total possible outcomes = C(52, 13) = 52! / (13! * (52-13)!) = 635,013,559,600
Favorable outcomes = C(13, 5) * C(39, 8) = (13! / (5! * (13-5)!) * (39! / (8! * (39-8)!) = 1287 * 1,559,495 = 2,009,563,845
Probability = Favorable outcomes / Total possible outcomes
Probability = 2,009,563,845 / 635,013,559,600 ≈ 0.00317
Therefore, the probability of getting exactly 5 spades in a hand of 13 cards is approximately 0.00317.
2. To determine whether it is more probable to get all 4 aces or all 13 cards of the same suit, we calculate the probabilities of each scenario.
Probability of getting all 4 aces:
Total possible outcomes = Number of ways to select 13 cards from a deck of 52 cards
Favorable outcomes = Number of ways to select all 4 aces and 9 non-ace cards from the remaining 48 cards.
Calculation:
Total possible outcomes = C(52, 13) = 635,013,559,600
Favorable outcomes = C(4, 4) * C(48, 9) = (4! / (4! * (4-4)!) * (48! / (9! * (48-9)!) = 1 * 18,424 = 18,424
Probability = Favorable outcomes / Total possible outcomes
Probability = 18,424 / 635,013,559,600 ≈ 0.000000029
Probability of getting all 13 cards of the same suit:
Total possible outcomes = Number of ways to select 13 cards from a deck of 52 cards
Favorable outcomes = Number of ways to select all 13 cards from one of the 4 suits.
Calculation:
Total possible outcomes = C(52, 13) = 635,013,559,600
Favorable outcomes = C(13, 13) * 4 = (13! / (13! * (13-13)!) * 4 = 1 * 4 = 4
Probability = Favorable outcomes / Total possible outcomes
Probability = 4 / 635,013,559,600 ≈ 0.0000000000063
Comparing the probabilities, it is more probable to get all 4 aces (approximately 0.000000029) than to get all 13 cards of the same suit (approximately 0.0000000000063).
Therefore, it is more probable to get all 4 aces than all 13 cards of the same suit.
To find the probability of a certain event, we need to divide the number of successful outcomes by the total number of possible outcomes. Let's solve each part of the problem:
a. Probability of getting exactly 5 spades:
First, let's consider the number of ways to choose 5 spades from the 13 available in the deck. This can be calculated using the combination formula, denoted as C(n, r), which calculates the number of possible combinations of 'r' elements chosen from a set of 'n' elements without regard to order. In this case, we need to find C(13, 5) since we're choosing 5 spades out of 13.
C(13, 5) = 13! / (5!(13-5)!) = 1287
Next, let's calculate the total number of possible hands. Since each of the four players is dealt 13 cards, the total number of possible hands is given by: C(52, 13) * C(39, 13) * C(26, 13) * C(13, 13) = 198,140,230,400
Therefore, the probability of getting exactly 5 spades is: 1287 / 198,140,230,400 ≈ 6.5 x 10^-9
b. Probability of getting all 4 aces:
Similarly, the number of ways to obtain all four aces is 1 (since there is only one set of four aces in the deck).
The total number of possible hands remains the same as calculated in part a, which is 198,140,230,400.
Therefore, the probability of getting all 4 aces is 1 / 198,140,230,400 ≈ 5.0 x 10^-12
c. Probability of getting all 13 cards of the same suit:
The number of ways to choose any specific suit from the four available is 1 (since we're interested in any suit). Once we choose the suit, we need to choose all 13 cards of that suit, which can be done in 1 way as well.
Hence, there is only 1 successful outcome for getting all 13 cards of the same suit.
The total number of possible hands remains the same at 198,140,230,400.
Therefore, the probability of getting all 13 cards of the same suit is 1 / 198,140,230,400 ≈ 5.0 x 10^-12
Comparing the probabilities:
We can see that the probabilities of getting all 4 aces and getting all 13 cards of the same suit are equal. Both probabilities are extremely low (on the order of 10^-12), indicating that both events are highly unlikely.
However, since the denominator is the same for both probabilities, the number of ways to obtain all 4 aces is only 1, while the number of ways to obtain all 13 cards of the same suit is 4 (one for each suit). Therefore, it is more probable to get all 13 cards of the same suit than to get all 4 aces.
let S be the event of being dealt a spade,
x the event of a non-spade
One such 5 spade hand could be
SSSSSXXXXXXXX
The prob of that is
(13/52*12/51*11/50*10/49*9/48)(39/47*38/46*37/45*36/44*35/43*34/33*33/32*32/31)
a quicker way would be to say
prob(exactly 5 spades)
= ( C(13,5)*C(39,8)/C(52,13)
= .12469
prob (4 aces)
= 4C(4,4)*C(48,9)/C(52,13) = appr .01056
prob(13 of same suit)
= 4/C(52,3) = very small
not even close!