PROBLEM SOLVING A drawer contains 12 white socks and 8 black...

PROBLEM SOLVING A drawer contains 12 white socks and 8 black socks. You randomly choose 1 sock and do not replace it. Then you randomly choose another sock. Find the probability that both events $A$ and $B$ will occur. (See Example $4 . )$ Event $A :$ The first sock is white. Event $B :$ The second sock is white.

PROBLEM SOLVING A drawer contains 12 white socks and 8 black socks. You randomly choose 1 sock and do not replace it. Then you randomly choose another sock. Find the probability that both events $A$ and $B$ will occur. (See Example $4 . )$
Event $A :$ The first sock is white.
Event $B :$ The second sock is white.

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Key Concepts

Dependent Events

Dependent events are events whose probabilities are affected by the occurrence of previous events. In scenarios like sampling without replacement, the outcome of the first event changes the sample space, thereby impacting the probability of subsequent events.

Multiplication Rule

The multiplication rule in probability states that the probability of two events occurring together is the product of the probability of the first event and the conditional probability of the second event given that the first has occurred. This rule is particularly important when dealing with sequential, dependent events.

Probability

Probability is the branch of mathematics that deals with measuring the likelihood of events occurring. It is calculated by taking the ratio of the number of favorable outcomes to the total number of possible outcomes and serves as a foundational concept in understanding random events.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. This concept is essential in situations where events are not independent, meaning the outcome of one event influences the outcome of another.

Transcript

00:07 Okay, for this question, we have a drawer that has 12 white socks and eight black socks.

00:14 We're going to choose one sock.

00:17 We are not going to replace it.

00:20 And then we're going to choose another sock.

00:24 Event a is going to be that our first sock is white.

00:27 And event b is going to be that our second sock is white.

00:31 We're going to find the probability of events a and b happening.

00:35 So we're going to choose one sock and then we're going to not replace it.

00:42 And because we do not replace it, the probability of event b is affected by the probability by event a happening.

00:54 So these events are called dependent events.

01:03 Okay.

01:04 And we have a formula to find the probability of dependent events.

01:07 It says that the probability of event a and b occurring is equal to the probability of event a multiplied by the probability of event b given that a has already happened.

01:26 Okay, so this is the probability of event b given that a has already happened.

01:33 So let's say we have our drawer here.

01:38 We have 12 white socks and we have eight black socks.

02:03 The first thing we need to do is find the probability of event a, which is the first sock is white.

02:11 So we're going to count how many white socks we have in our drawer.

02:16 That would be 12...

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