Suppose you are dealt 5 cards from a standard 52 -card deck....

Suppose you are dealt 5 cards from a standard 52 -card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52 card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4 th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?

Suppose you are dealt 5 cards from a standard 52 -card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions:
(a) How many ways can 5 cards be selected from a 52 card deck?
(b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck?
(c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4 th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards?
(d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?

Key Concepts

Probability Computation for Card Hands

This concept explains that the probability of drawing a specific type of hand is the ratio of the number of favorable hands to the total number of possible hands. In the context of a 5-card hand from a 52-card deck, it involves dividing the count of hands that satisfy the given conditions by the total combinations available.

Counting with Restrictions

Counting with restrictions involves determining the number of outcomes when specific limitations are imposed, such as ensuring the remaining two cards are of different ranks from each other and from the triple. This requires careful selection and combination of items while excluding unwanted cases.

Multiplication Rule

The multiplication rule states that if there are several independent choices or stages in a process, the total number of outcomes is the product of the number of ways each stage can occur. This is used to combine the counts from different selections, such as choosing a rank for the triple and then selecting the remaining cards from different ranks.

Combination Formula

This concept involves calculating the number of ways to choose k items from n without regard to order. It is essential for determining the total number of possible hands (for example, 52 choose 5 for a standard deck) as well as for selecting specific groups of cards, like choosing 3 cards out of 4 for a particular rank.

Transcript

00:01 This question is asking us to calculate the probability of getting a certain five card hand.

00:05 Specifically, we're trying to get three of a kind when you're just being dealt that five card hand from a standard 52 card deck.

00:13 And the four parts of this question are really just leading us through the big answer because this is a pretty complicated probability.

00:22 So looking at part a, we want to know how many ways can five cards be selected from a 52 card deck.

00:29 So for part a, i've got 52 total cards and i'm choosing a group of five.

00:36 I'm using a combination here because the order of these cards doesn't matter.

00:41 You're just getting a group of five cards when you're dealt that hand.

00:45 And so the combination, remember, uses the first, the big total is the first number.

00:51 So 52 cards in total.

00:53 And we are essentially counting groups of size five.

00:56 So that's where the five is coming from.

00:58 So this number comes out to be 2 ,598 ,960.

01:05 So that's how many total five card hands there are.

01:07 That's going to be the denominator of our eventual probability calculation.

01:14 Now, in part b, we have to be able to figure out how many ways you can get three of the same card selected from the deck.

01:22 So you want to imagine that you're kind of rifling through the deck of cards and you are picking your.

01:28 Group of three of one kind.

01:31 And so the way, that's how i like to think about it.

01:33 And so imagine, you know, first you have to decide, um, which, which kind do you want? which three of a kind? you want three aces.

01:40 Do you want three kings? do you want three fives? and the number of choices you have for that is 13 because there are 13 different kinds.

01:48 You can also phrase this as 13 choose one because of the 13 different kinds.

01:53 You're just picking initially one of them to be the, the three of a kinds that you're going to want.

01:59 So once i've, let's say, decided on queens, i want three queens.

02:03 Now i have to figure out, okay, how many groups of three queens can be chosen? because there are four total...

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