In this tutorial, you're going to learn about "And" probability. Specifically you will focus on:
"And" means that you're going to do A and B. Events A and B both have to be the case. And they're going to be independent.
There's different "And" probability formulas for independent events versus dependent events. This tutorial is going to deal with independent events.
Take a look.
Suppose a coin is flipped and this spinner is spun and apparently A and D are a lot more common than B or C.
Since the outcome of the coin flip doesn't have any influence on the probabilities for the spinner, these events are called independent. Spinning the spinner wouldn't have any effect on the probability of the coin coming up heads. And knowing if the coin came up heads doesn't affect the probabilities for the spinner.
You can create a tree diagram to show all the outcomes.
There are two outcomes for the quarter and there are four outcomes for the spinner, meaning there are eight total outcomes. But they're not all equally likely.
The heads and tails each have a one-half probability. That the A, B, C and D don't all have the same probability. If you look closely at the spinner, A is the biggest sector, followed by D, and then B and C.
In fact, the truth is that they are in this ratio. 35% A, 25% B, 10% C, 30% D.
Since it doesn't matter whether you got heads the first time or tails the first time-- the probabilities will remain the same-- You can apply them down below too.
What's the probability that you get both heads and sector D?
Suppose that you did this experiment 1,000 times.
You would expect about 500 heads and 500 tails, if everything went according to your expectation. If the second group went according to your expectation, 35% of those 500 would say A. 25% of those 500 would say B. 10% of those 500 would say C. And 30% of those 500 would say D.
Ultimately, you would end up with 175 of the 1,000 that went heads A, 125 of the 1,000 that went heads B. Ultimately what you’re looking for is heads D, heads and D. 150 of the 1,000 that you did. When you simplify 150 out of 1,000 you get 0.15.
Is there another way to obtain 0.15? Look at the path that you traveled to get here. You went from heads down to D. You had those probabilities of 1/2 and 0.3. The probability of H and D was 0.15. Hopefully, what you're seeing is that these two values multiplied give us 0.15. What you have is a multiplication for independent events.
The "And" probability, the probability of events A and B both happening, is the product of their individual probabilities. In this case, it was 1/2 for heads and 0.3 for D. Multiply them and you got 0.15. This is an important rule to remember.
There's a special multiplication rule for finding "And" probability, which is also called joint probability for two independent events. The probability that they both occur is equal to the product of their individual probabilities. There will be a different multiplication rule if the events aren't independent.
Good luck.
Source: This work adapted from Sophia Author Jonathan Osters.
A way to approximate probability based on trials of chance experiments that mimic the real-life trials. Also known as the "Special Multiplication Rule".