Conditional Probability (College Board AP® Statistics): Study Guide

Look out for the phrase "given that" in an exam question. This phrase normally indicates that a conditional probability is needed.

There are 10 same-sized counters in a bag, 4 of them are red. A counter is chosen at random and it is not replaced. A second counter is then chosen at random.

What is the probability that the second counter is red, given that the first counter is red?

Answer:

Initially, there are 10 counters: 4 are red and 6 are not red

It is given that a red counter is chosen first and not replaced

This means there are now 9 counters remaining: 3 are red and 6 are not red

Find the probability that the second counter is red given that the first counter is red by dividing the number of remaining red counters by the total number of remaining counters

$P\left(\mathrm{second}\mathrm{is}\mathrm{red}|\mathrm{first}\mathrm{is}\mathrm{red}\right)=\frac{3}{9}$

The probability can be simplified

Given that the first counter is red, the probability that the second counter is red is $\frac{1}{3}$

A common mistake that students make is using $P\left(A\right)$ instead of $P\left(A\cap B\right)$ on the numerator of the fraction. If $B$ has already occurred then you essentially want to ignore any outcomes that are not in $B$.

In a school, a particular group of students are required to choose one math subject (Calculus or Statistics) and one language (French or Spanish).

17% of students chose Calculus and French, whereas 23% of students chose Calculus and Spanish. 25% of students chose Statistics and French and 35% of students chose Statistics and Spanish.

What is the probability that a randomly selected student chose Statistics, given that they also chose Spanish?

Answer:

The required probability is $P\left(\mathrm{Statistics}|\mathrm{Spanish}\right)$

The conditional probability formula is $P\left(\mathrm{Statistics}|\mathrm{Spanish}\right)=\frac{P\left(\mathrm{Statistics}\mathrm{and}\mathrm{Spanish}\right)}{P\left(\mathrm{Spanish}\right)}$

$P\left(\mathrm{Statistics}\mathrm{and}\mathrm{Spanish}\right)$ is given in the question but $P\left(\mathrm{Spanish}\right)$ is not

Find $P\left(\mathrm{Spanish}\right)$ by adding together the probabilities of the combinations involving Spanish

$\begin{array}{rcl}P\left(\mathrm{Spanish}\right)& =& P\left(\mathrm{Calculus}\mathrm{and}\mathrm{Spanish}\right)+P\left(\mathrm{Statistics}\mathrm{and}\mathrm{Spanish}\right)\\ & =& 0.23+0.35\\ & =& 0.58\end{array}$

Substitute the relevant probabilities into the conditional probability formula

$\begin{array}{rcl}P\left(\mathrm{Statistics}|\mathrm{Spanish}\right)& =& \frac{P\left(\mathrm{Statistics}\mathrm{and}\mathrm{Spanish}\right)}{P\left(\mathrm{Spanish}\right)}\\ & =& \frac{0.35}{0.58}\\ & =& \frac{35}{58}\end{array}$

Given that a randomly selected student chose Spanish, the probability that they also chose Statistics is $\frac{35}{58}$ (roughly 0.603)