Select three different five card combinations or five-card hands from your favorite card game that utilizes a standard 52-card deck containing four suits (clubs, hearts, diamonds, and spades), with each suit containing 13 cards with numbers 2-10 and face cards ace, king, queen, and jack. Then, do the following.
Using the concept of dependent probabilities, determine the odds that you would draw these hands (card combinations) directly from a deck of cards.
Determine the probability that you would not draw these hands (card combinations) directly from a deck of cards.
This is what I got for the first part and I feel it's wrong. Didn't work on the second part yet.
Statistics - Shasha, Thursday, May 25, 2017 at 10:40pm
When looking at the possibilities of drawing 5 cards from a deck of 52 there are going to be 2,598,960 possibilities. Now if we are looking to pull 2 kings, a heart, and 2 10’s, it would be 4/52, 3/51, 4/50, 4/49, 3/48 = (4*3*4*4*3/52*51*50*49*48) = The probabilities these cards would be pulled in a five-card combination is 576/ 311,875,200 =1/541,450 or 1.86E-06
Here is the answer I got, am I on the right track?
The probability of the heart in your example would be 13/50 if the king or ace did not include a heart.
You have still not included probability that either/both a king or ace will be a heart. Thus the probability of the heart in your next equations would be either 12/50 or 11/50.
Either-or probabilities are found by adding the individual probabilities.
Add the probabilities of the three equations.
Yes, you are on the right track with your calculations for the first part of the question. However, let's go through the calculations step by step to make sure we have the correct answer.
To determine the odds of drawing a specific five-card combination directly from a deck of cards, we need to consider the number of favorable outcomes (the number of ways we can get the desired combination) and the number of total outcomes (the total number of possible combinations).
Let's take the example of drawing 2 kings, a heart, and 2 tens.
Number of favorable outcomes:
- There are 4 kings in the deck, and we want to draw 2 of them, so we have 4 choose 2 (4C2) possibilities.
- There are 13 hearts in the deck, and we want to draw 1 of them, so we have 13C1 possibilities.
- There are 4 tens in the deck, and we want to draw 2 of them, so we have 4C2 possibilities.
- The remaining two cards can be any of the 52 cards that are not kings, hearts, or tens, so we have 52 - 4 - 13 - 4 - 4 = 27C2 possibilities.
Multiplying these possibilities together, we get: (4C2) * (13C1) * (4C2) * (27C2) = (6) * (13) * (6) * (27*26/2) = 1,716 * 13 * 6 * 351 = 16,090,344.
Number of total outcomes:
- There are 52 cards in the deck, and we want to draw 5 of them, so we have 52 choose 5 (52C5) possibilities.
Calculating the probability:
The probability is the ratio of the number of favorable outcomes to the number of total outcomes.
So, the probability of drawing the specific five-card combination is: 16,090,344 / (52C5) = 16,090,344 / 2,598,960 ≈ 0.0062069 or approximately 0.62%.
For the second part of the question, to determine the probability that you would not draw these hands directly from the deck of cards, you need to subtract the probability of drawing the specific five-card combination from 1.
So, the probability that you would not draw these hands is: 1 - 0.0062069 = 0.9937931 or approximately 99.38%.
I hope this helps! Let me know if you have any further questions.