🎲 probability

1. **Problem statement:** Given that $P(A|B) + P(B|A) = 0.8$ and $\frac{1}{P(B)} + \frac{1}{P(A)} = 4$, find $P(B \cap A)$. 2. **Recall the conditional probability formulas:**

1. **Problem statement:** An urn contains 5 balls. Two balls are drawn and both are white. We want to find the probability that all 5 balls in the urn are white. 2. **Understanding

1. **State the problem:** We have 7 red counters and some blue counters in a bag. The total number of counters is unknown. We know there are more blue counters than red counters.

1. **Problem statement:** We are given the probability that a car is stolen overnight in an unsafe area is $p=0.1$. There are $n=12$ cars parked. We want to find probabilities for

1. **State the problem:** We are given that the probability a car is stolen overnight in an unsafe area is 10%, or $p=0.1$. There are 12 cars parked on the street. We want to find

1. **Problem Statement:** We are given a probability distribution for the number of defective bulbs $X$ produced in a day:

1. **Problem statement:** A box contains 50 marbles: 13 blue, 10 red, and 27 yellow. A marble is taken without replacement. Find:

1. The problem is to find the complement of an event $q$ in probability or set theory. 2. The complement of an event $q$, denoted as $q^c$ or $\overline{q}$, consists of all outcom

1. The problem asks for the probability of picking the correct answer at random from a set of possible answers. 2. The formula for probability when choosing one correct answer from

1. **Problem statement:** You roll a fair 6-sided die 17 times. Find the probability of getting exactly 5 even numbers and exactly 3 fives. 2. **Understanding the problem:** Each r

1. **Problem statement:** We have a set of customer accounts with an error rate of 5% (probability of error $p=0.05$). The auditor samples $n$ accounts randomly.

1. **State the problem:** We want to generate random numbers in the interval $[0,1]$ with 2-digit accuracy, meaning numbers like $0.00, 0.01, 0.02, \ldots, 0.99, 1.00$. 2. **Unders

1. **State the problem:** We want to transform random numbers uniformly distributed on the interval $[0,1]$ into random numbers uniformly distributed on the interval $[-148,1128]$.

1. **Problem Statement:** We have two jars: Jar A with 6 green and 4 red balls, Jar B with 2 green and 8 red balls. One ball is drawn from each jar. Given that the balls drawn have

1. **State the problem:** We have a set $S = \{1,2,3,4,5,6,7,8,9,10\}$ and we want to find the probability that a number selected at random from $S$ is odd. 2. **Formula for probab

1. The user requested a tree diagram, which is a visual representation used to show all possible outcomes of an event or series of events. 2. Tree diagrams are not mathematical pro

1. **Stating the problem:** We have a survey of 1000 students at University ABC, with 750 boys and 250 girls. Among smokers, 22.9% are boys and 4.5% are girls. We need to construct

1. **Problem:** Given the pdf \(f(x) = 0.075x + 0.2\) for \(3 \leq x \leq 5\) and 0 otherwise, verify it is a valid pdf and calculate probabilities. 2. **Step 1: Verify total area

1. **Problem:** Given the probability density function (pdf) $$f(x) = \begin{cases} 0.075x + 0.2 & 3 \leq x \leq 5 \\ 0 & \text{otherwise} \end{cases}$$

1. **Problem statement:** A committee consists of 7 men and 4 women. A sub-committee of 6 is chosen at random. Find the probability that the sub-committee contains exactly 2 women.

1. **Problem Statement:** An experiment consists of asking 3 persons at random if they wash their clothes using detergent. (a) List the elements of a sample space $S$ using the let