Q: A bag contains 5 red balls and 7 blue balls. If you draw two balls without replacement, what is the probability that both are red?

$$ \frac{5}{12} \times \frac{4}{11} $$

Question 1

Which of the following statements best describes the difference between independent and dependent events in probability?

Accepted Answer

Independent events are those where the occurrence of one event does not affect the probability of the other, while dependent events are those where it does.

Question 2

A bag contains 5 red balls and 7 blue balls. If you draw two balls without replacement, what is the probability that both are red?

Accepted Answer

$$ \frac{5}{12} 	imes \frac{4}{11} $$

Question 3

Which of the following is NOT a requirement for a valid probability distribution?

Accepted Answer

Each probability must be greater than 1.

Question 4

What is the formula for the expected value $$E(X) of a $$discrete random variable $$X$$ with probability distribution $$P(X)$$?

Accepted Answer

$$ E(X) = \sum X \cdot P(X) $$

Question 5

A fair six-sided die is rolled 4 times. What is the probability that at least one roll shows a 6?

Accepted Answer

$$ 1 - \left( \frac{5}{6} \$$right)^4$$ $$

Question 6

Which formula would you use to calculate the number of ways to arrange 12 unique books on a shelf?

Accepted Answer

$$ 12! $$

Question 7

Which of the following is the correct formula for the mean of a binomial distribution with parameters $$n$$ and $$p$$?

Accepted Answer

$$ \mu = n \cdot p $$

Question 8

Which of the following is NOT a requirement for a binomial distribution?

Accepted Answer

Each trial must have more than two possible outcomes.

Question 9

A hospital records the number of patients arriving per hour. If the average is 3 patients per hour, which distribution best models the probability of exactly 5 patients arriving in one hour?

Accepted Answer

Poisson distribution

Question 10

What is the formula for the probability of exactly $$k$$ successes in $$n$$ binomial trials with probability $$p of $$success?

Accepted Answer

$$ P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$ $$

Question 11

Which of the following best describes a uniform distribution?

Accepted Answer

All outcomes are equally likely within a given interval.

Question 12

If $$P(A) = 0.3$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are disjoint events, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$ 0.8 $$

Question 13

Which of the following is the correct formula for the z-score of a value $$x in a $$normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$ z = \frac{x - \mu}{\sigma} $$

Question 14

Which of the following statistics is an unbiased estimator of the population mean?

Accepted Answer

Sample mean $$\bar{x}$$

Question 15

A normal distribution has mean $$\mu = 50$$ and standard deviation $$\sigma = 5. $$According to the Range Rule of Thumb, what values are considered significantly high?

Accepted Answer

$$ x \geq 60 $$

Question 16

Which formula would you use to calculate the probability of the complement of event $$A$$?

Accepted Answer

$$ P(A') = 1 - P(A) $$

Question 17

In a binomial experiment with $$n = 10$$ and $$p = 0.2$$, what is the standard deviation?

Accepted Answer

$$ \sigma = \sqrt{n p (1-p)} $$

Question 18

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 19

Which of the following is the correct formula for the probability of exactly $$k$$ events in a Poisson distribution with mean $$\lambda$$?

Accepted Answer

$$ P(X = k) = \frac{\lambda^k e$$^{-\lambda}$$}{k!} $$

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