# Tagged: conditional probability

## Problem 755

Let $X$ and $Y$ be geometric random variables with parameter $p$, with $0 \leq p \leq 1$. Assume that $X$ and $Y$ are independent.

Let $n$ be an integer greater than $1$. Let $k$ be a natural number with $k\leq n$. Then prove the formula
$P(X=k \mid X + Y = n) = \frac{1}{n-1}.$

## Problem 754

We have a stick of a unit length. Two points on the stick will be selected randomly (uniformly along the length of the stick) and independently. Then we break the stick at these two points so that we get three pieces of the stick. What is the probability that these three pieces form a triangle?

## Problem 752

Alice tossed a fair coin until a head occurred. Then Bob tossed the coin until a head occurred. Suppose that the total number of tosses for Alice and Bob was $7$.

Assuming that each toss is independent of each other, what is the probability that Alice tossed the coin exactly three times?

## 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?

## 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?

## 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?

## Problem 737

Two fair coins are tossed. Given that at least one of them lands heads, what is the conditional probability that the first coin lands heads?

## 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.

## Problem 732

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.

## 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?

## Problem 730

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 heads?

## Problem 729

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?

## Problem 728

A fair six-sided die is rolled.

(1) What is the conditional probability that the die lands on a prime number given the die lands on an odd number?

(2) What is the conditional probability that the die lands on 1 given the die lands on a prime number?