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 subsequent tests. If a gold ring failed to pass one of the tests, what is the probability that it failed the second test?

Let $F$ be the event that a gold ring fails one of the three tests. Let $F_2$ be the event that it fails the second test. Then what we need to compute is the conditional probability
\[P(F_2 \mid F) = \frac{P(F_2 \cap F)}{P(F)}.\]

The numerator is
\[P(F_2 \cap F) = P(F_2) = 0.9 \cdot 0.15.\]
(A gold ring passes the first test with probability $0.9$ and fails the second test with probability $1-0.85=0.15$.)

The complement $F^c$ of $F$ is the event that a gold ring passes all the tests. Thus
\[P(F) = 1- P(F^c) = 1 – 0.9 \cdot 0.85 \cdot 0.8.\]
It follows that the desired probability is
\begin{align*}
P(F_2 \mid F) &= \frac{0.9 \cdot 0.15}{1 – 0.9 \cdot 0.85 \cdot 0.8}
= \frac{135}{388} \approx 0.348
\end{align*}

Therefore, given that a gold ring failed to pass one of the tests, the probability that it failed the second test is about 34.8 %.

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 […]

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 […]

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 […]

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 […]

Find the Conditional Probability About Math Exam Experiment
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 […]

Complement of Independent Events are Independent
Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$.
Prove that $E$ and $F^c$ are independent as well.
Solution.
Note that $E\cap F$ and $E \cap F^c$ are disjoint and $E = (E \cap F) \cup (E \cap F^c)$. It follows that
\[P(E) = P(E \cap F) + P(E […]