Any time you want to know the chance of two events happening together, you can use the multiplication rule of probability.
Independent events:
P(A and B) = P(A) x P(B)
Dependent events:
P(A and B) = P(A) x P(B | A)
...where P(B | A) is the probability of event B given that event A happened.
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Video transcript:
What’s the chance of rolling snake eyes? What’s the chance of flipping heads three times in a row?
When calculating the probability of two or more events happening together, we can use the multiplication rule of probability.
We’ll start with independent events where one event’s outcome has no effect on the other event’s outcome. For example, what’s the probability of rolling snake eyes? Each roll is an independent event because the value on one die has no influence on the value of the second die.
The multiplication rule of probability says that the probability of two events A and B happening together is the probability of event A multiplied by the probability of event B - in this case, the probability of rolling a 1 on the first die, multiplied by the probability of rolling a 1 on the second die. This is the case if you’re rolling the dice together or one at a time. The probability of rolling a 1 on a die is one out of six, so the probability of rolling a 1 on both dice is ⅙ times ⅙. Across all 36 possible rolls of two dice, one of them is snake eyes.
Let’s look at a second example - what’s the probability of flipping heads three times in a row? Well, it’s the probability of the first flip landing heads, multiplied by the probability of the second flip landing heads, multiplied by the probability of the third flip landing heads. ½ x ½ x ½ = ⅛.
What if the events are dependent? What if the second event’s probability is based on the outcome of the first event? In this case, the probability of the events happening together is a little more complicated - the probability of event A happening multiplied by the probability of event B happening given that event A happened. For example, what’s the probability that you’ll draw an ace, hold onto it, and then draw a king? In this case, we’ll start with the probability of drawing an ace: four out of 52 cards are aces. Then, we need to calculate the probability of drawing a king, given that we’ve already drawn an ace, which is different than the probability of drawing a king from a full deck. There are four kings left in the deck, and there are 51 cards remaining since we’re holding onto an ace. We multiply these two probabilities together and we get a probability of 16/2652, about a 1 in 167 chance.
In summary, if you want to know the likelihood of event A and event B happening, you can use the multiplication rule of probability. Make sure to identify whether the events are independent or dependent and adjust your calculation accordingly.
