2.1.3.2.5 - Conditional Probability

Conditional Probability

The probability of one event occurring given that it is known that a second event has occurred. This is communicated using the symbol $$\mid$$ which is read as "given."

For example, $$P(A\mid B)$$ is read as "Probability of A given B."

The two-way table below displays the World Campus enrollment from Fall 2015 in terms of level (undergraduate and graduate) and biological sex. Given that an individual is an undergraduate student, what is the probability that the student is female?

Female Male Total
Total 6027 6215 12242

We know that the individual is an undergraduate student so we will only look at the 7242 undergraduate students. Of those 7242 undergraduate students, 3814 were female.

$$P(F \mid U) = \frac{3814}{7242}=0.527$$

Conditional probabilities can also be computed using the following formulas. Note that these two formulas are identical, but A and B are switched. Again, if the contingency table is available it is usually most efficient to take the appropriate values from the table, as shown above, as opposed to using these formulas.

Conditional Probability of A Given B
$$P(A\mid B)=\dfrac{P(A \: \cap\: B)}{P(B)}$$
Conditional Probability of B Given A
$$P(B\mid A)=\dfrac{P(A \: \cap\: B)}{P(A)}$$

In a large class, the probability of randomly selecting a woman is .60. The probability of randomly selecting a student who is a woman and who earned an A is .20. If you randomly select a student who is a woman, what is the probability that she earned an A?

Here, we will call event A earning a grade of A. We will call event B being a woman. We want to know the probability of earning an A given that the student is a female. In other words, $$P(A\mid B)$$

We are given that $$P(A \: \cap\: B)=.20$$ and $$P(B)=.60$$

$$P(A\mid B) =\frac{P(A \: \cap\: B)}{P(B)}=\frac{.20}{.60}=.333$$

Given that a randomly selected student is a woman, there is a 33.3% chance that she earned an A.

Example: Clubs Section

In a standard 52-card deck, 25% of cards are clubs, 50% of cards are black, and 25% of cards are black clubs. What is the probability that a randomly selected card is a club given that it is a black card?

We are given that $$P(club)=.25$$, $$P(black)=.50$$, and  $$P(club \: \cap\: black)=.25$$

$$P(club\mid black)=\frac{P(club \: \cap\: black)}{P(black)}=\frac{.25}{.50}=.50$$

Given that a randomly selected card is black, there is a 50% chance that it's a club.

Independent Events Written as Conditional Probabilities Section

If events A and B are independent then $$P(A) = P(A \mid B)$$. In other words, whether or not event B occurs does not change the probability of event A occurring.