Conditional Probability with Venn Diagrams

1. Conditional Probability With Venn Diagrams

A Venn diagram is a useful tool to visualize conditional probability.

EXAMPLE

What is the probability of getting an even on a roulette wheel? Like the example above, this question is not a conditional probability yet because the question is simply about the probability that the number is even. To find the answer, count up all of the even sectors.

hint

Notice that zero and double zero don't count as even. The evens on the roulette wheel are in two categories: even numbers that are in black, and even numbers that are in red. Eighteen of the 38 numbers on the roulette wheel are even.

However, what if we want to know the probability that the sector is even, given that the sector is also black?

This is a conditional probability statement. We can create a Venn diagram to show the relationship between these two characteristics of even and black. Ignore any of the sectors that are neither black nor even, and the ones that are only even without being black. Some of the black numbers are also even.

Of the 18 numbers that are black on the roulette wheel, 10 selections are both even and black. So, the probability is 10 even black sectors out of 18 total black sectors.

We could also use the conditional probability formula, which states we need to find the probability of a sector being both even and black, which is 10 out of 38, and divide by the probability of a sector being black, which is 18 out of 38.

In the example above, we show all the individual members of each group, but usually Venn diagrams summarize this by only indicating the total number of individuals in that group, as in the example below.

try it

The Venn diagram represents the protein powder flavor preferences of 100 bodybuilders.

What is the probability that a bodybuilder likes chocolate protein powder, given we know they like vanilla?

Apply the conditional probability formula.

hint

To calculate P(vanilla), be sure to count both the 28 who like vanilla only and the 22 who like both flavors.
(28/100 + 22/100 = 0.50)