A corporation was concerned with the basic educational...

A corporation was concerned with the basic educational skills of its workers and decided to offer a selected group of them separate classes in reading and practical mathematics. Of these workers, $40 \%$ signed up for the reading classes and $50 \%$ for the practical mathematics classes. Of those signing up for the reading classes $30 \%$ signed up for the mathematics classes. a. What is the probability that a randomly selected worker signed up for both classes? b. What is the probability that a randomly selected worker who signed up for the mathematics classes also signed up for the reading classes? c. What is the probability that a randomly chosen worker signed up for at least one of these two classes? d. Are the events "signs up for the reading classes" and "signs up for the mathematics classes" statistically independent?

A corporation was concerned with the basic educational skills of its workers and decided to offer a selected group of them separate classes in reading and practical mathematics. Of these workers, $40 \%$ signed up for the reading classes and $50 \%$ for the practical mathematics classes. Of those signing up for the reading classes $30 \%$ signed up for the mathematics classes.
a. What is the probability that a randomly selected worker signed up for both classes?
b. What is the probability that a randomly selected worker who signed up for the mathematics classes also signed up for the reading classes?
c. What is the probability that a randomly chosen worker signed up for at least one of these two classes?
d. Are the events "signs up for the reading classes" and "signs up for the mathematics classes" statistically independent?

Key Concepts

Basic Probability

Basic probability is the study of likelihood of events occurring, quantifying uncertainty with a number between 0 and 1. In this context, it involves determining the probabilities that workers participate in different classes without considering relationships between the events initially.

Conditional Probability

Conditional probability refers to the probability of one event occurring given that another event has occurred. It is crucial in cases where the occurrence of one event affects or restricts the outcome of another, allowing us to calculate probabilities in a more informed context.

Joint Probability

Joint probability is the probability of two events happening simultaneously. It is calculated by considering the intersection of both events, often through the multiplication of a conditional probability and the probability of the conditioning event. This concept is key when assessing the likelihood of combined participation in both classes.

Union of Events

The union of events involves combining the probabilities of two events to determine the chance that at least one occurs. This concept requires adjustments for any overlap between the events, ensuring that the shared outcomes are not counted twice.

Statistical Independence

Statistical independence is a condition where the occurrence of one event does not affect the probability of the occurrence of another event. Determining whether events are statistically independent involves comparing the joint probability to the product of the individual probabilities.

Transcript

00:01 Now, here in this problem, we're told that a corporation was concerned with the basic educational skills of its workers.

00:05 Now, of the workers, 40 signed up for the reading classes, and so this means the probability of reading is 0 .40.

00:15 50 % signed up for the math, and so the probability that they signed up for the math is equal to 0 .50.

00:24 Now, of those signing up for the reading, 30 % signed up for the mathematics classes.

00:29 What that means is that the probability of math given reading is 0 .30.

00:35 If we know that they are in reading, there's a 30 % chance that they sign up for the math.

00:40 On a, we want the probability a randomly selected worker signed up for both classes.

00:45 And we want the probability then of r and, and the probability of r and m is equal to the probability of m given r times the probability of r.

01:04 This is just our conditional probability law.

01:09 Probability of m given r is 0 .30.

01:13 Probability of r probability of reading is 0 .40 and so we're going to take 0 .3 times 0 .4 and that gives us 0 .12.

01:27 And so the probability of both of them would be 0 .12.

01:33 On b, we want the probability that randomly selected workers signed up for the math also signed up for the reading.

01:39 So in other words, we want the probability of reading given math...

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