Find the Conditional Probability About Math Exam Experiment

Problem 740

A researcher conducted the following experiment. Students were grouped into two groups. The students in the first group had more than 6 hours of sleep and took a math exam. The students in the second group had less than 6 hours of sleep and took the same math exam.

The pass rate of the first group was twice as big as the second group. Suppose that $60\%$ of the students were in the first group. What is the probability that a randomly selected student belongs to the first group if the student passed the exam?

Let $E$ be the event that a student passes the exam.
Let $G_i$ be the event that a student belongs to the group $i$ for $i= 1, 2$. Then the desired probability is $P(G_1 \mid E)$. ($G_1$ is the first group and $G_2$ is the second group.)

Using these notation, the pass rate of the first group is expressed as $P(E \mid G_1)$. Similarly, the pass rate of the second group is $P(E \mid G_2)$. By assumption, the pass rate of the first group is twice as big as the second group. Hence, we have
\[P(E \mid G_1) = 2\ P(E \mid G_2).\]

The required probability that a randomly selected student belongs to the first group given that the student passes the exam is expressed as $P(G_1 \mid E)$.

Using the above equality and Bayes’ rule, we get
\begin{align*}
&P(G_1 \mid E)\\[6pt]
&= \frac{P(G_1) P(E \mid G_1)}{P(E)} & \text{(by Bayes’ rule)}\\[6pt]
&= \frac{P(G_1)P(E \mid G_1)}{P(E \mid G_1)P(G_1) + P(E \mid G_2)P(G_2)} & \text{(rule of total probability)}\\[6pt]
&= \frac{P(G_1)\cdot 2P(E \mid G_2)}{2P(E \mid G_2)P(G_1) + P(E \mid G_2)P(G_2)} \\[6pt]
&= \frac{P(G_1)\cdot 2}{2P(G_1) + P(G_2)}\\[6pt]
&= \frac{0.6 \cdot 2}{2\cdot 0.6 + 0.4}\\[6pt]
&= \frac{3}{4}.
\end{align*}

Therefore, the required probability is $P(G_1 \mid E) = \frac{3}{4}$.

Overall Fraction of Defective Smartphones of Three Factories
A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively.
Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively. Determine the overall fraction of […]

If a Smartphone is Defective, Which Factory Made It?
A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively. Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively.
If a smartphone of this model is […]

Probability of Having Lung Cancer For Smokers
Let $C$ be the event that a randomly chosen person has lung cancer. Let $S$ be the event of a person being a smoker.
Suppose that 10% of the population has lung cancer and 20% of the population are smokers. Also, suppose that we know that 70% of all people who have lung cancer […]

Independent Events of Playing Cards
A card is chosen randomly from a deck of the standard 52 playing cards.
Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart.
Prove or disprove that the events $E$ and $F$ are independent.
Definition of Independence
Events […]

Jewelry Company Quality Test Failure Probability
A jewelry company requires for its products to pass three tests before they are sold at stores. For gold rings, 90 % passes the first test, 85 % passes the second test, and 80 % passes the third test. If a product fails any test, the product is thrown away and it will not take the […]

Pick Two Balls from a Box, What is the Probability Both are Red?
There are three blue balls and two red balls in a box.
When we randomly pick two balls out of the box without replacement, what is the probability that both of the balls are red?
Solution.
Let $R_1$ be the event that the first ball is red and $R_2$ be the event that the […]

What is the Probability that All Coins Land Heads When Four Coins are Tossed If…?
Four fair coins are tossed.
(1) What is the probability that all coins land heads?
(2) What is the probability that all coins land heads if the first coin is heads?
(3) What is the probability that all coins land heads if at least one coin lands […]