Conditional Probability: idea and how to use it

What is Conditional Probability?

Conditional probability measures the likelihood (probability) of one event (A) given that another event (B) has occurred. In other words, what’s the chance of A also happening if we know B has happened? Let \(P[A]\) denote the probability of event A.

The conditional probability of A given event B is denoted as \(P[A|B] \). The symbol “\(|\)” means “given”. It can be obtained from $$P[A|B] = \frac{P[A \cap B]}{P[B]}$$

Why Do We Care About Conditional Probability?

If we forget about the formal definition and formula for a moment and think about our everyday life scenarios, we would see that we encounter the concept of conditional probability all the time.

• Weather: What is the likelihood of rain today, given that it is currently cloudy? (Should I bring an umbrella?)
• Vaccine efficacy: The chance of contracting a virus may be much less when the patients take the vaccine, compared with when they do not take the vaccine.
• Genetic disorder: What is the chance a child will inherit a specific trait, given that one parent carries the gene? 
• Auto insurance rate: Is the chance that young people have an accident higher if they are younger than 30? Therefore, they should be charged at a higher rate.

The definition and formula enable us to model and quantify the likelihood of one event given that another event has occurred, such as the situations mentioned above.

Examples

Although real-life examples are more fun and exciting, they are also big projects that require background knowledge irrelevant to probability. To keep us focused on the topic of conditional probability, we use textbook examples. The first example is a variation of the famous Two-Child Problem. The initial formulation of the question dates back to at least 1959.

Example 1

A coin is flipped twice. Assuming that all four points in the sample space \(S = \{ (H,H),(H, T), (T,H),(T,T) \}\) are equally likely, what is the conditional probability that both flips land on heads, given that at least one flip lands on heads?

Let \( A= \{(H,H)\} \) be the event that both flips land on heads. Let \(B = \{ (H,H),(H, T),(T,H) \}\) be the event that at least one flip lands on heads. Then \( A \cap B = \{(H,H)\} \). Moreover, \( P[A\cap B] = \frac 14\) and \(P[B] = \frac 34\). The conditional probability that both flips land on heads, given that at least one flip lands on heads is $$ P[A|B] = \frac{ P[A\cap B]}{P[B]} = \frac{1/4}{3/4} = \frac 13$$

Example 2

Joe is 80 percent certain that his missing key is in one of the two pockets of his hanging jacket, being 40 percent certain it is in the left-hand pocket and 40 percent certain it is in the right-hand pocket. If a search of the left-hand pocket does not find the key, what is the conditional probability that it is in the other pocket?

Let \(L\) be the event that the key is in the left-hand pocket of the jacket, and \(R\) be the event that it is in the right-hand pocket. Since the event, \(R \cap L’\), that the key is not in the left pocket and is in the right pocket is the same as \(R\), we obtain the desired conditional probability \(P[R | L’]\) as follows:

\(\begin{aligned}
P[R | L’] &= \frac{P[R \cap L’]}{P[ L’]} \\
&=\frac{P[R ]}{1 – P[ L]} \\
&= \frac{0.4}{1-0.4} \\
&= \frac{2}{3}
\end{aligned}\)

More Advanced Examples

We have more examples selected from sample questions for the Actuarial Probability Exam.