A single card is drawn from a standard deck. Let A be the...

A single card is drawn from a standard deck. Let A be the event that "the card is a face card" (a jack, a queen, or a king), $\mathbf{B}$ is a "red card," and $\mathrm{C}$ is "the card is a heart." Determine whether the following pairs of events are independent or dependent: a. $\quad A$ and $B$ b. $\mathrm{A}$ and $\mathrm{C}$ c. $\quad B$ and $C$

A single card is drawn from a standard deck. Let A be the event that "the card is a face card" (a jack, a queen, or a king), $\mathbf{B}$ is a "red card," and $\mathrm{C}$ is "the card is a heart." Determine whether the following pairs of events are independent or dependent:
a. $\quad A$ and $B$
b. $\mathrm{A}$ and $\mathrm{C}$
c. $\quad B$ and $C$

Key Concepts

Counting Techniques in Uniform Sample Spaces

Probability problems involving finite uniform sample spaces often rely on counting methods, where the total number of outcomes and favorable outcomes are determined by combinatorial analysis. In such contexts, every outcome is equally likely, and calculating probabilities involves counting the number of outcomes that satisfy the event conditions relative to the total possible outcomes. This technique is fundamental when working with structured experiments like card draws or dice rolls.

Event Intersection and Its Calculation

The concept of intersection refers to the event where two (or more) events occur simultaneously. To determine whether events are independent or dependent, it is necessary to compute the probability of their intersection. By comparing this probability to the product of the single events’ probabilities, one can assess independence. Accuracy in computing these intersections is essential in a variety of probability-based problems.

Independence of Events

In probability theory, two events are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other. This is mathematically expressed as P(A ? B) = P(A) × P(B), meaning that the joint probability of both events occurring is the product of their individual probabilities. This concept is crucial when analyzing whether events influence each other or occur completely by chance.

Transcript

00:01 Problem 113 is all about playing cards and we're given three events.

00:07 So a is the event that the card is a jack, a queen or a kick.

00:14 B is the event that it's red and c is the event that it's a heart.

00:21 Now we are given a pair of these events and ask to determine whether they're independent or not.

00:27 So a is asking us about a and b.

00:33 So does the chance of one change depending on the other? so if we look at a, let's say we drew a jack, there is still a two and four chance, which is one and two, of it being red.

00:52 Because half the cards are red and half the cards are black.

00:57 And same way the other way around.

00:59 If we drew a red card it doesn't affect the chance of being a jack because there are two red jacks and there are two black jacks.

01:06 So because of that you can say these are independent.

01:12 Part b is asking us about a and c.

01:18 So if we draw a heart does that change the likelihood of being a jack queen or king and vice versa? so if you draw a heart, three of them, so you've now got a 3 and 13 because there are 13 hearts are in the jack queen kick.

01:35 If you drew a spade you'd have the same chance, same with the club, same with the diamond.

01:40 So the fact that you've drawn a heart doesn't change your chance of getting a jack, queen or king...

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