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Compound Probability: Definition & Examples

Instructor: Sharon Linde
What is compound probability? How is it calculated? When is it appropriate to use this technique? The answers to these and other related questions await you in the following lesson.

Compound Probability in Real Life

Let's say you're a huge soccer fan. Your favorite player is Lionel Messi, while a friend's favorite player is Cristiano Ronaldo. She bets you $100.00 Ronaldo will score in his next league game, and Messi will not. You say: 'I'll take that bet!'

Is this a good idea or just another example of you making a bad sports call? Let's figure it out.

Defining Compound Probability

Compound probability may sound complicated, but it really just refers to the probability of two or more independent events both happening. The term, independent events, means that the outcome of one event has no effect on the outcome of another event.

The actual calculation is quite straightforward: You multiply the two independent probabilities together. The compound probability equation looks like this:

P(A and B) = P(A) x P(B)

Calculating Compound Probability

Earlier in the lesson, you backed Messi because he's your favorite player, but who's more likely to win the bet: you or your friend? To find the answer to this question, you'll have to do some research and find out how many goals Messi and Renaldo scored in each game.

When you've finished with your research, you see that Messi scored 43 goals in 38 games played, while Ronaldo scored 48 goals in 35 games played. Uh oh! You may have made that bet a bit too quickly because it looks like Ronaldo scores more goals than Messi. Luckily you remember that you didn't bet on the number of goals but on the likelihood of them scoring in a particular game. Whew! Now that you're not sweating bullets anymore, we need to find out what those numbers look like.

To figure out who wins the bet, you first have to read the data and plug the numbers into the equation. Let's review what you learned during your research:

  • Messi scored in 25 out of 38 games.
  • Ronaldo scored in 25 out of 35 games.

Since Messi scored in 25 out of 38 games, his probability of scoring in the next game is 25/38. As Ronaldo failed to score in 10 out of 35 games, we'll use the following steps to solve the equation:

1.) P(A and B) = P(A) x P(B)

2.) P(Messi scores and Ronaldo does not) = P(Messi scores) x P(Ronaldo does not score)

3.) (25/38) x (10/35)

4.) 0.6579 x 0.2857

5.) 0.1879 x 100

6.) 18.79%

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