Five cards are drawn from an ordinary deck without replacement. Find the probability of getting
a. All red cards
b. All diamonds
c. All aces
a)26C5/52C5
b)13C5/52C5
c)48/52C5
52C5=2598960
26C5=65780
13C5=1287
48=48
a. 26/52 * 25/51 * 24/50 * 23/49 * 22/48 = ?
b. 13/52 * 12/51 * 11/50 *10/49 * 9/48 = ?
c. 4/52 * 3/51 * 2/50 * 1/49 = ? It doesn't matter what the fifth card is.
a. Ah, the joy of drawing all red cards! The probability can be calculated as follows: since there are 26 red cards in a standard deck, and we're drawing 5 cards in total without replacement, the probability is (26/52) * (25/51) * (24/50) * (23/49) * (22/48). Simplifying this, we get 5/16. So, the probability of drawing all red cards is 5/16.
b. Diamonds are forever, right? Well, the probability of drawing all diamonds can be found by dividing the number of ways to draw 5 diamonds (13 choose 5) by the total number of possible 5-card combinations (52 choose 5). After doing some math magic, we find that the probability is 1287/2598960. So, the probability of drawing all diamonds is 1287/2598960.
c. Aces are the masters of the deck! To find the probability of drawing all aces, we simply divide the number of ways to draw all 4 aces (4 choose 4) by the total number of possible 5-card combinations (52 choose 5). After some calculations, we get the probability as 1/2598960. Wow, those aces are quite elusive! So, the probability of drawing all aces is 1/2598960.
To find the probability of getting specific combinations of cards from a deck, it's important to understand the concept of probability and then apply it systematically.
In this case, we have an ordinary deck of 52 cards, which consists of 26 red cards (13 hearts and 13 diamonds) and 4 aces (one in each suit).
Let's go through each scenario step-by-step:
a. All red cards:
First, we need to determine the total number of ways we can choose 5 cards out of the 52-card deck. This can be done using the combination formula, which is nCr = n! / (r!(n-r)!), where n is the total number of cards and r is the number of cards we want to choose. In this case, we have n = 52 and r = 5.
So, the total number of ways to choose 5 cards from a deck of 52 cards is 52! / (5!(52-5)!) = 52! / (5!47!).
Next, we need to determine how many ways we can choose 5 red cards from the 26 available red cards. This can be calculated using the same combination formula, but this time n = 26 and r = 5.
The number of ways to choose 5 red cards from 26 red cards is 26! / (5!(26-5)!) = 26! / (5!21!).
Finally, to find the probability, we divide the number of favorable outcomes (choosing 5 red cards) by the number of possible outcomes (choosing any 5 cards):
Probability of getting all red cards = (number of ways to choose 5 red cards) / (number of ways to choose 5 cards from the deck)
= (26! / (5!21!)) / (52! / (5!47!))
b. All diamonds:
For this scenario, we need to determine how many ways we can choose 5 diamonds from the 13 available diamonds in the deck. Using the combination formula, n = 13 and r = 5.
The number of ways to choose 5 diamonds from 13 diamonds is 13! / (5!(13-5)!) = 13! / (5!8!).
Probability of getting all diamonds = (number of ways to choose 5 diamonds) / (number of ways to choose 5 cards from the deck)
= (13! / (5!8!)) / (52! / (5!47!))
c. All aces:
In this scenario, we need to determine how many ways we can choose all 4 aces from the deck. Since there is only one ace in each suit, there is only one favorable outcome.
Probability of getting all aces = (number of ways to choose 4 aces) / (number of ways to choose 5 cards from the deck)
= 1 / (52! / (5!47!))
Please note that in all these probability calculations, we assume that the cards are drawn without replacement, which means that once a card is drawn, it is not put back into the deck before drawing the next card.