Question 1

Suppose you toss a fair coin three times. Which of the following is a valid probability distribution for the number of heads observed?

Accepted Answer

$$P(0) = 0.125,\ P(1) = 0.375,\ P(2) = 0.375,\ P(3) = 0.125$$

Question 2

A probability distribution is given as follows: $$P(X=0) = 0.3$$, $$P(X=1) = 0.5$$, $$P(X=2) = 0.3. Is $$this a valid probability distribution?

Accepted Answer

No, because the probabilities sum to 1.1, which is greater than 1.

Question 3

Given a random variable $$X$$ with the following probability distribution: $$P(X=1) = 0.2$$, $$P(X=2) = 0.5$$, $$P(X=3) = 0.3$$, what is the probability that $$X is $$greater than or equal to 2?

Accepted Answer

$$0.8$$

Question 4

A game costs $$5 to play. You win 20 if $$you roll a 6 on a fair six-sided die, and nothing otherwise. What is the expected value of your net gain per game?

Accepted Answer

$$-1.67$$

Question 5

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

Accepted Answer

The probability of success changes from trial to trial.

Question 6

A fair die is rolled 10 times. What is the probability of getting exactly 2 sixes? (Use the binomial formula: $$P(X=k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$, where p$$ = \frac{1}{6}$$)

Accepted Answer

$$\binom{10}{2} \left(\frac{1}{6}\$$right)^2$$ \left(\frac{5}{6}\$$right)^8$$

Question 7

If the mean of a normal distribution is $$\mu = 50$$ and the standard deviation is $$\sigma = 5$$, what is the $$z-$$score for a value of $$60$$?

Accepted Answer

$$z = 2$$

Question 8

A normal random variable $$X$$ has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the probability that $$X is $$greater than $$130$$? (Use the $$z-$$score formula: $$z = \frac{x - \mu}{\sigma})$$

Accepted Answer

About $$0.0668$$

Question 9

Which of the following best describes a significantly high value in a normal distribution?

Accepted Answer

A value that is more than two standard deviations above the mean.

Question 10

A binomial experiment consists of 8 trials with probability of success $$p = 0.3. $$What is the expected number of successes?

Accepted Answer

$$2.4$$

Question 11

If a normal distribution has mean $$\mu = 70$$ and standard deviation $$\sigma = 8$$, what value separates the top 10% of the distribution? (Use the $$z-$$score for the 90th percentile, $$z \approx 1.28)$$

Accepted Answer

$$80.24$$

Question 12

A sample mean is found to be 2.5 standard deviations above the population mean. According to the 'two standard deviations' rule, how would you classify this sample mean?

Accepted Answer

Significantly high

Question 13

Which of the following scenarios is NOT a binomial experiment?

Accepted Answer

Measuring the time it takes for a chemical reaction to complete.

Question 14

A normal distribution has $$\mu = 120$$ and $$\sigma = 10. $$What is the probability that a randomly selected value is less than $$110$$? (Use the $$z-$$score formula and standard normal table.)

Accepted Answer

About $$0.1587$$

Question 15

In a binomial experiment with $$n = 12$$ and $$p = 0.4$$, what is the probability of getting exactly 5 successes? (Use the binomial formula.)

Accepted Answer

$$\binom{12}{5} (0.4)^5$ (0.6)^7$

Question 16

A random variable $$X$$ has the following probability distribution: $$P(X=0) = 0.1$$, $$P(X=1) = 0.3$$, $$P(X=2) = 0.4$$, $$P(X=3) = 0.2. $$What is the expected value $$E(X)$$?

Accepted Answer

$$1.7$$

Question 17

Which of the following is the correct formula for the standard deviation of a binomial distribution with $$n$$ trials and probability $$p of $$success?

Accepted Answer

$$\sqrt{np(1-p)}$$

Question 18

A medical test for a rare disease has a 99% accuracy rate. If 1000 people are tested and 1% actually have the disease, what is the expected number of true positives?

Accepted Answer

$$10$$

Question 19

If the probability of a rare event is $$0.002$$, what is the probability of observing at least one occurrence in 500 independent trials? (Use the complement rule: $$P(	ext{at least one}) = 1 - (1-p)^n)$$

Accepted Answer

About $$0.632$$

Probability Distributions and Binomial Probability: Study Notes for Statistics Exam Prep

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