Drawing Cards Suppose that you draw 3 cards without replacemen...

Question

Drawing Cards Suppose that you draw 3 cards without replacement from a standard 52-card deck. What is the probability that all 3 cards are aces?

Accepted Answer

Welcome back, everyone. In this problem, 4 cards are drawn at random without replacement from a standard 52 card deck. What is the probability that all four cards are from the same suit? A says 0.1006. B 0.6010, C 0.0106, and D 0.0116. Now, in this problem, since we are asked to find the probability that all 4 cards are drawn from the same suit, that means we'll need to figure out the total number of ways to select 4 cards from a standard 52 card deck and the number of ways to select 4 cards from a single suit, then to select it from 4 suits, right? Now, let's start by using our combination formula to help us since we are drawing without replacement. Recall that by that formula it says that n chooser. Is going to be equal to n factorial divided by r factorial multiplied by N minus R factorial. So in that case it means then that if we have 52 cards in this case n would be 52, and we're choosing 4, so R would be 4, we'll have 52 choose 4, which will be 52 factorial divided by 4 factorial, multiplied by 52 minus 4 or 48 factorial. When we go ahead and solve here, we should get this to be equal to. 270,725 So this is the the the total number of ways to select 4 cards from a 52 card deck. Next, we need to calculate the number of ways to select 4 cards from the same suit. Now, since there are 4 suits, that is, hearts, diamonds, clubs, and spades, we choose 1 suit, select 4 cards from the 13 cards in that suit, and then multiply that answer by 4. So in this case, that means 1st 13 choose 4 then is going to be equal to 13 factorial, multiplied by 4 factorial, multiplied by 13 minus 4 or 9 factorial. When we calculate that equals 715. So this means then that the total favorable outcomes. Since there are 4 suits. Is going to be equal to 4 multiplied by the total number of ways to choose one suit, which is 13 choose 4 or 715, and that is 2860. So now that we have the total possible outcomes, to choose 4 cards from the standard 270,725, and the total favorable outcomes to choose four cards from a single suit, 2860, then we can find the probability because remember, probability is equal to the number of favorable outcomes divided by the total outcomes. So the probability we choose all 4 cards from the same suit. Is going to be equal to. 2860 divided by 270,725 and that is going to be approximately 0.0106. What does this tell us? Well, the probability of all four cards drawn out from the same suit would be 0.0106. And if we look back at our answer choices, that means C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Drawing Cards Suppose that you draw 3 cards without replacement from a standard 52-card deck. What is the probability that all 3 cards are aces?

Key Concepts

Probability Without Replacement

Counting Favorable Outcomes

Multiplication Rule for Dependent Events

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