Five cards are drawn at random without replacement from and ordinary deck of 52. What is the probability all 5 cards are from the same suit?
1 of the 4 suits ---> C(4,1) = 4
5 from one of those suits = C(13,5) ---> 1287
5 from the whole deck ---> C(52,5) = 2598960
so prob = 4(1287)/2598960
= 33/16660 = .00198
I'm confused as to how you're getting these numbers... Could you possible break it down a little farther please? I don't understand how you get from C(52,5)=2598960...Thank you so much for helping!
Have you learned the difference between permutations and combinations?
in combinations the order does not matter.
(if I hold 5 cards in my hand, the order in which they were dealt to me does not matter)
C(13,5) means ..
how many groups of 5 can I form from 13
It is short form for 13!/(5!8!)
= 13x12x11x10x...x2x1/(8x7x6x..x2x1)(5x4x3x2x1))
= 13x12x11x10x9/(5x4x3x2x1) = 154440/120 = 1287
On your calculator there is a button labeled
nCr
so enter
13
2nd F
nCr
5
=
You should get 1287
253/9996
To find the probability of drawing all five cards from the same suit, we need to first determine the total number of favorable outcomes (i.e., drawing all cards from the same suit) and the total number of possible outcomes (i.e., any combination of five cards drawn from the deck).
Step 1: Determine the total number of favorable outcomes.
We have four different suits in a deck of 52 cards: hearts, diamonds, clubs, and spades. Each suit contains 13 cards. So, the number of favorable outcomes is:
Number of favorable outcomes = 4 (number of suits) * 1 (since we want all five cards to be from the same suit) = 4
Step 2: Determine the total number of possible outcomes.
In the first draw, we can choose any of the 52 cards from the deck. After the first card is drawn, there are only 51 cards remaining. For the second card, we can choose any of the remaining 51 cards. This process continues until all five cards are drawn. Therefore, the total number of possible outcomes is:
Number of possible outcomes = 52 * 51 * 50 * 49 * 48
Step 3: Calculate the probability.
The probability of an event is given by the ratio of the number of favorable outcomes to the number of possible outcomes. Thus, the probability of drawing all five cards from the same suit is:
Probability = Number of favorable outcomes / Number of possible outcomes
Probability = 4 / (52 * 51 * 50 * 49 * 48)
Now, we can calculate the probability using a calculator or mathematical software.
Note: The probability could also be expressed as a decimal or percentage.