Unit 4: Probability, Random Variables and Distributions

If [math] and [math], and [math] and [math] are independent, then [math]

The expected value of a discrete random variable is:

A binomial distribution requires all of the following conditions EXCEPT:

If [math], what is the mean of [math]?

The addition rule for probability states that [math]

The geometric distribution models:

If [math] and [math] are independent random variables with [math] and [math], then [math]

Conditional probability [math] is calculated as:

A continuous random variable differs from a discrete random variable in that:

Two events A and B are mutually exclusive. If P(A) = 0.3 and P(B) = 0.5, what is P(A or B)?

In a class, 60% of students pass the midterm, 80% pass the final, and 50% pass both. What is the probability a student passes the final given they passed the midterm?

Events A and B are independent. P(A) = 0.4 and P(B) = 0.3. What is P(A and B)?

P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.2. What is P(A or B)?

A bag contains 3 red and 5 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?

A disease affects 1% of a population. A test for the disease is 95% accurate (both sensitivity and specificity). If a person tests positive, what is the approximate probability they actually have the disease?

A die is rolled 3 times. What is the probability of getting at least one six?

A bag has 4 red and 6 blue marbles. A marble is drawn, replaced, and another is drawn. What is P(both red)?

The law of large numbers states that:

What is the probability that a randomly selected student plays both sports and music?

Given that a student passed, what is the probability that they studied?

Are the events "being female" and "preferring brand A" independent?

What is the expected profit per game for the carnival?

What is the probability that exactly 3 of the 5 selected items are defective?

What is the probability that the first free throw made is on the 3rd attempt?

What are the mean and standard deviation of the total cost X + Y?

What proportion of bags weigh less than 498 grams?

A medical test for a disease has a 95% true positive rate (sensitivity) and a 3% false positive rate. The disease affects 1 in 200 people. If a randomly selected person tests positive, what is the approximate probability that they actually have the disease?

At a factory, machine A produces 60% of items and machine B produces 40%. Defect rates are 2% for A and 5% for B. An item selected at random is defective. What is the probability it came from machine B?

A game involves rolling two fair dice and summing the results. Let X = the sum. What is P(X ≥ 10)?

A discrete random variable X has the following distribution: P(X=1) = 0.2, P(X=2) = 0.3, P(X=3) = 0.3, P(X=5) = 0.2. What is the expected value E(X) and the variance Var(X)?