MAT prob&stat1 discrete probability distributions

Question 1

What is the binomial distribution used to model?

Answer 1

Situations with multiple trials, each having two possible outcomes: success or failure. It uses the binomial coefficient to calculate probabilities.

Question 2

What are the conditions for using the binomial distribution?

Answer 2

• n independent trials
• Two possible outcomes (success and failure)
• Fixed probabilities for success (p) and failure (1-p)
• The random variable X represents the number of successes.

Question 3

How is the probability of r successes in n trials calculated?

Answer 3

Using the formula:
P(X = r) = ⁿCᵣ * p^r * (1-p)^(n-r)

Question 4

What is the notation for a random variable following a binomial distribution?

Answer 4

X ~ B(n, p), where n is the number of trials and p is the probability of success.

Question 5

What is the expectation and variance of a binomial random variable X ~ B(n, p)?

Answer 5

Expectation: E(X) = np
Variance: Var(X) = np(1-p)

Question 6

When tossing a coin 20 times, what is the binomial distribution of the number of heads?

Answer 6

X ~ B(20, 0.5), where n = 20 and p = 0.5.

Question 7

How do you find the probability P(X ≥ r) using binomial tables?

Answer 7

Use the complement:
P(X ≥ r) = 1 - P(X ≤ r-1)

Question 8

What is the probability of getting at least one success in a binomial distribution?

Answer 8

P(X ≥ 1) = 1 - P(X = 0)

Question 9

In a class of 30 students, if the pass rate for a test is 60%, what is the distribution of the number of failures?

Answer 9

X ~ B(30, 0.4), where p = 0.4 (probability of failure)

Question 10

How do you determine the number of trials required for a specific probability threshold?

Answer 10

Use the complement rule and trial and improvement or logarithms to solve inequalities involving binomial probabilities.

Question 11

If a flight has a 90% on-time rate, what is the probability all 4 flights are on time?

Answer 11

P(X = 4) = (0.9)^4 = 0.656

Question 12

What is the most likely outcome in a binomial distribution?

Answer 12

The value of X closest to the mean np, which also has the highest probability.

Question 13

What is the geometric distribution used to model?

Answer 13

Situations where trials are repeated until the first success occurs, with the number of trials required being the random variable X.

Question 14

What is the probability formula for success on the n-th trial in a geometric distribution?

Answer 14

P(X = n) = (1 - p)^(n-1) * p
where p is the probability of success and (1 - p) is the probability of failure.

Question 15

What is the notation for a random variable following a geometric distribution?

Answer 15

X ~ Geo(p), where p is the probability of success.

Question 16

What are the conditions for using the geometric distribution?

Answer 16

• Trials are independent.
• Each trial has two outcomes (success or failure).
• Probability of success (p) is constant for all trials.
• Trials continue until the first success occurs.

Question 17

How does the geometric distribution differ from the binomial distribution?

Answer 17

The geometric distribution has an unknown number of trials and stops at the first success, while the binomial distribution has a fixed number of trials.

Question 18

What are the mean and variance of a geometric distribution?

Answer 18

Mean: μ = 1 / p
Variance: σ^2 = (1 - p) / p^2

Question 19

If X ~ Geo(0.3), what is P(X = 5)?

Answer 19

P(X = 5) = (1 - 0.3)^4 * 0.3 = 0.072

Question 20

If X ~ Geo(0.3), what is P(X > 8)?

Answer 20

P(X > 8) = (1 - 0.3)^8 = 0.0576

Question 21

If X ~ Geo(0.3), what is P(X < 4)?

Answer 21

P(X < 4) = 1 - P(X > 3) = 1 - (1 - 0.3)^3 = 0.657

Question 22

Hassan spins a spinner with p = 1/6 until he gets a six. What are the mean and variance of his spins?

Answer 22

Mean: μ = 1 / p = 6
Variance: σ^2 = (1 - p) / p^2 = 30