In this tutorial, you're going to learn about the general "and" probability formula for dependent events. This tutorial will cover:
"And" probability for independent events, which is discussed in its own tutorial, has a special multiplication rule that is a general multiplication rule.
General Multiplication Rule
The probability that two events both occur is the probability of the first event times the conditional probability that the second event occurs, given that the first already has.
Suppose there's a famous television show "Go For Broke.” On the show, a person gets to roll a die to determine which jar they're going to draw a colored chip from. If the chip is green, they will win a prize, and if the chip is red, they won’t win anything.
If a person rolls an even number, they can select from the A jar that contains 7 green chips and 13 red chips. Choosing from that jar gives you a pretty good probability to win the prize.
If a person rolls odd, they select from the B jar, which contains 5 green chips and 25 red chips. That jar doesn’t give you a very good likelihood of winning.
The probability of picking a red chip depends on the die roll. That means that these two events are dependent. The probability of picking a red chip increases if you roll an odd number versus if you roll an even number.
Let's look at it in a tree diagram:
There's a 1/2 probability of rolling an odd and a 1/2 probability of rolling an even. But that's where the similarities end:
The circled number is the probability of selecting a green if you've already rolled an odd. This is a conditional probability: the probability of green given odd.
What if you’re multiplying the probability of an odd times the probability of green, given odd? This requires the general probability formula, the "and" probability formula for dependent events.
Knowing the outcome of the first event, A, doesn't change the probability of the second event, B. Using symbolic conditional probability notation, you can say the probability that B occurs given that we know that A happened is the same as just the probability of B, whether or not we knew A was happening.
And so what does that mean within the formula? That means the probability of A and B is equal to this by the general multiplication rule. But the probability of B given A is the same as the probability of B.
This formula looks familiar. That's the special multiplication rule for independent events.
The special multiplication rule is a special version of the general multiplication rule.
Conditional probability is the joint probability of two events occurring together, either concurrently or consecutively, is equal to the probability of the first event times the probability of the second event given that the first event occurred. It's best to think about this like you're going down the branches of a tree diagram. The "And" probability general formula works for both independent and dependent events.
Thanks and good luck!
Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS
The probability that two events both occur is the probability of the first event times the conditional probability that the second event occurs, given that the first already has.