Each card in a standard deck of 52 cards belongs to one of...

Each card in a standard deck of 52 cards belongs to one of four different suits: hearts, diamonds, spades, or clubs. There are 13 cards in each suit. Consider a scenario in which you draw five cards from the deck, one at a time, and record only the suit to which each card drawn belongs. (a) Describe the sample space. (b) What is the probability that the set of five cards you draw consists of two spades, one heart, one diamond, and one club (drawn in any order)? (c) What is the probability that exactly two of the five cards you draw are from the same suit?

Each card in a standard deck of 52 cards belongs to one of four different suits: hearts, diamonds, spades, or clubs. There are 13 cards in each suit. Consider a scenario in which you draw five cards from the deck, one at a time, and record only the suit to which each card drawn belongs.
(a) Describe the sample space.
(b) What is the probability that the set of five cards you draw consists of two spades, one heart, one diamond, and one club (drawn in any order)?
(c) What is the probability that exactly two of the five cards you draw are from the same suit?

Key Concepts

Combinatorial Analysis (Combinations and Permutations)

Combinatorial analysis provides methods for counting the number of ways items can be selected or arranged. Combinations are used when the order does not matter and permutations are used when the order does matter. These techniques are crucial for evaluating the number of favorable outcomes in probability problems involving specific arrangements or selections.

Probability Calculation

Probability calculation is the process of determining how likely an event is by using the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. This approach is fundamental to solving problems in probability, where both counting the possible outcomes and identifying those that satisfy the event's criteria are key.

Sample Space

The sample space in probability is the complete set of all possible outcomes of a random experiment. It serves as the foundational basis for any probabilistic analysis, ensuring that every outcome is accounted for, so that probabilities can be accurately assigned to events, which are subsets of the sample space.

Multiplication Principle (Fundamental Counting Principle)

This principle states that if a task can be divided into a sequence of independent steps, where each step can be performed in a fixed number of ways, then the total number of outcomes is the product of the number of ways each step can occur. It is essential in counting the total number of outcomes, particularly in multi-stage experiments like sequential card draws.

Transcript

00:01 Okay, so this problem's got a multi -part solution to it.

00:03 So part a just wants us to describe the sample space of drawing five cards from a standard deck of cards and only recording the suits.

00:11 So to describe it, the first thing we can do is we can get hearts and get all five to be hearts.

00:21 Okay.

00:23 The second one, we could get four hearts in a club, third, three spades, two hearts.

00:29 So basically, we're drawing cards without replacement.

00:32 So each heart that we grabbed would be less likely to get the next one.

00:38 And so our sample space could be fairly large.

00:44 And so the sample space size is going to, i'm going to call it s.

00:52 That's going to be we have a choice of 52 for the first slot, 51, 509, 48.

01:05 And that number is relatively large is around 312 million so it's a fairly large number of the size of this sample space so 311 875 200 that is the size of our sample space part b of what's the probability that the set of five cards you draw consist of two spades, one heart, one diamond, and one club, and draw it in any order.

01:59 Okay, so to calculate that particular case, we have out of the 311, 875, 200 ways, actually, okay, we do want to draw this in any order.

02:23 Okay, so the number of hearts to choose from on the first draw.

02:29 Let's just say the first draw has to be the most limiting is that we have two spades...

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