Category: Probability

Interchangeability of Limits and Probability of Increasing or Decreasing Sequence of Events

Problem 744

A sequence of events $\{E_n\}_{n \geq 1}$ is said to be increasing if it satisfies the ascending condition
\[E_1 \subset E_2 \subset \cdots \subset E_n \subset \cdots.\] Also, a sequence $\{E_n\}_{n \geq 1}$ is called decreasing if it satisfies the descending condition
\[E_1 \supset E_2 \supset \cdots \supset E_n \supset \cdots.\]

When $\{E_n\}_{n \geq 1}$ is an increasing sequence, we define a new event denoted by $\lim_{n \to \infty} E_n$ by
\[\lim_{n \to \infty} E_n := \bigcup_{n=1}^{\infty} E_n.\]

Also, when $\{E_n\}_{n \geq 1}$ is a decreasing sequence, we define a new event denoted by $\lim_{n \to \infty} E_n$ by
\[\lim_{n \to \infty} E_n := \bigcap_{n=1}^{\infty} E_n.\]

(1) Suppose that $\{E_n\}_{n \geq 1}$ is an increasing sequence of events. Then prove the equality of probabilities
\[\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right).\] Hence, the limit and the probability are interchangeable.

(2) Suppose that $\{E_n\}_{n \geq 1}$ is a decreasing sequence of events. Then prove the equality of probabilities
\[\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right). \]

Read solution

LoadingAdd to solve later

Successful Probability of a Communication Network Diagram

Problem 742

Consider the network diagram in the figure. The diagram consists of five links and each of them fails to communicate with probability $p$. Answer the following questions about this network.

(1) Determine the probability that there exists at least one path from A to B where every link on the path functions without errors. Express the answer in term of $p$.

(2) Assume that exactly one link has failed. In this case, what is the probability that there is a successful path from A to B, that is, each link on the path has not fail.

reliability of communication network

Read solution

LoadingAdd to solve later

Lower and Upper Bounds of the Probability of the Intersection of Two Events

Problem 741

Let $A, B$ be events with probabilities $P(A)=2/5$, $P(B)=5/6$, respectively. Find the best lower and upper bound of the probability $P(A \cap B)$ of the intersection $A \cap B$. Namely, find real numbers $a, b$ such that
\[a \leq P(A \cap B) \leq b\] and $P(A \cap B)$ could take any values between $a$ and $b$.

Read solution

LoadingAdd to solve later

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?

Read solution

LoadingAdd to solve later

What is the Probability that Selected Coin was Two-Headed?

Problem 739

There are three coins in a box. The first coin is two-headed. The second one is a fair coin. The third one is a biased coin that comes up heads $75\%$ of the time. When one of the three coins was picked at random from the box and tossed, it landed heads.

What is the probability that the selected coin was the two-headed coin?

Read solution

LoadingAdd to solve later

If a Smartphone is Defective, Which Factory Made It?

Problem 738

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 found out to be detective, what is the probability that this smartphone was manufactured in factory C?

Read solution

LoadingAdd to solve later

Probability of Having Lung Cancer For Smokers

Problem 736

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 are smokers.

Then determine the probability of a person having lung cancer given that the person is a smoker.

Read solution

LoadingAdd to solve later

Overall Fraction of Defective Smartphones of Three Factories

Problem 735

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 defective smartphones of this model.

Read solution

LoadingAdd to solve later

Jewelry Company Quality Test Failure Probability

Problem 731

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?

Read solution

LoadingAdd to solve later

Probability Problems about Two Dice

Problem 727

Two fair and distinguishable six-sided dice are rolled.

(1) What is the probability that the sum of the upturned faces will equal $5$?

(2) What is the probability that the outcome of the second die is strictly greater than the first die?

Read solution

LoadingAdd to solve later