Probability and Statistics > Probability > Multiplication Rule Probability

## Multiplication rule probability (General)

The **multiplication rule** is a way to find the probability of two events happening at the same time (this is also one of the AP Statistics formulas). There are two multiplication rules. The general multiplication rule formula is: P(A ∩ B) = P(A) P(B|A) and the specific multiplication rule is P(A and B) = P(A) * P(B). P(B|A) means “the probability of A happening given that B has occurred”.

## Multiplication Rule Probability (Specific)

The **specific multiplication rule**, P(A and B) = P(A) * P(B), is *only* valid if the two events are independent. In other words, it only works if one event does not change the probability of the other event.

Examples of independent events :

- Owning a cat and getting a weekly paycheck.
- Finding a parking space and having a coin for the meter.
- Buying a book and then buying a coffee.

### Multiplication Rule Probability: Using the Specific Rule

Using the specific multiplication rule formula is very straightforward. Just multiply the probability of the first event by the second. For example, if the probability of event A is 2/9 and the probability of event B is 3/9 then the probability of both events happening at the same time is (2/9)*(3/9) = 6/81 = 2/27.

### Multiplication Rule Probability: Using the General Rule

This rule can be used for any event (they can be independent or dependent events). You still have to multiply two numbers, but first you have to use a little logic to figure out the second probability before multiplying.

**Sample problem:** a bag contains 6 black marbles and 4 blue marbles. Two marbles are drawn from the bag, without replacement. What is the probability that both marbles are blue?

Step 1: Label your events A and B. Let A be the event that marble 1 is blue and let B be the event that marble 2 is blue.

Step 2: Figure out the probability of A. There are ten marbles in the bag, so the probability of drawing a blue marble is 4/10.

Step 3: Figure out the probability of B. There are nine marbles in the bag, so the probability of choosing a blue marble (P B|)A is 3/9.

Step 4: Multiply Step 2 and 3 together: (4/10)*(3/9) = 2/15.

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## References

Beyer, W. H. CRC Standard Mathematical Tables, 31st ed. Boca Raton, FL: CRC Press, pp. 536 and 571, 2002.

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.

Wheelan, C. (2014). Naked Statistics. W. W. Norton & Company