Cards In Exercises 59-62, you are dealt a hand of five cards f...

Question

Cards In Exercises 59-62, you are dealt a hand of five cards from a standard deck of 52 playing cards.

59. Find the probability of being dealt two clubs and one of each of the other three suits.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A committee of 4 is to be formed from 5 men and 7 women. What is the probability that the committee will have exactly 2 men and 2 women? Awesome. So it appears for this particular problem, we're asked to determine what the probability is that the committee will have exactly 2 men and 2 women. Awesome. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. So A is 0.3182, B is 0.4167, C is 0.4242, and finally D is 0.4000. Awesome. So first off, in order to solve for this particular problem, we need to recall and use combinations and basic probability concepts. So with that in mind, we need to determine the number of ways to choose 2 men from 5. So the number of ways to choose 2 men from 5 is we need to have 5 choose 2, which is a binomial coefficient. Which a binomial coefficient is just a fancy way of writing the combination formula, and when you perform the combination formula for 5 chooses 2, you will determine that it's equal to 10. And then our second step is we need to determine the number of ways to choose two women from 7. So this would be 7 chooses 2, and when you perform that calculation, you will find that it's equal to 21, noting once again we're using the combination formula for both step 1 and step 2. So finally, for step 3, we need to recall a note that the committee will have exactly 2 men and 2 women. So we need to multiply these values to get the num the total number of favorable outcomes, so we need to take 10 multiplied by 21, which will give us an answer of 210. Our fourth step is we need to find the total number of outcomes, so we need to determine the total number of ways to choose any 4 people from 12. So that will mean we need to take 12, choose 4, so 12 choose 4, so when we apply the combination formula for 12 choose 4, we will find that it's equal to 495, so 495. So now, our 5th and final step is we need to determine the probability, and the probability P is equal to the number of favorable outcomes which you'll call FO for short, divided by the total number of outcomes which you'll call TNO. So once again, the probability. is equal to the number of favorable outcomes divided by the total number of outcomes, which is equal to when we plug in our known variables, 210 divided by 495. Which is equal to when we round to 4 decimal places, 0.4242, and that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer without a doubt has to be multiple choice answer letter C, which once again is 0.4242. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Cards In Exercises 59-62, you are dealt a hand of five cards from a standard deck of 52 playing cards.
59. Find the probability of being dealt two clubs and one of each of the other three suits.

Key Concepts

Probability

Combinatorics

Deck of Cards

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