Question 1

Which of the following is a property of a probability distribution for a discrete random variable?

Accepted Answer

The sum of all probabilities must be 1.

Question 2

Suppose the waiting time for a bus is uniformly distributed between 0 and 15 minutes. What is the probability that a person waits less than 12.5 minutes? Use the formula for the uniform distribution: $$P(a \leq X \leq b) = \frac{b-a}{B-A}$$ where $$A$$ and $$B$$ are the minimum and maximum values.

Accepted Answer

$$P = \frac{12.5-0}{15-0} = \frac{12.5}{15} = 0.833$$

Question 3

A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.25$$, $$P(X=1)=0.50$$, $$P(X=2)=0.25. $$What is the expected value (mean) of $$X$$? Use $$\mu = \sum x \cdot P(x).$$

Accepted Answer

$$\mu = 0 \cdot 0.25 + 1 \cdot 0.50 + 2 \cdot 0.25 = 1.0$$

Question 4

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Counting the number of heads in 5 coin tosses.

Question 5

A die is tossed 3 times. What is the probability of getting no fives? (Let success be rolling a five, $$p=\frac{1}{6}$$, $$n=3)$$

Accepted Answer

$$P(X=0) = (\frac{5}{6})^3$ = 0.5787$$

Question 6

Which of the following tables represents a valid probability distribution?

Accepted Answer

x: 8, 7, 5; P(x): 0.3, 0.2, 0.5

Question 7

A probability distribution has $$\mu = 2$$ and $$\$$sigma^2$$ = 3. $$What is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{3} \approx 1.73$$

Question 8

Jessica makes 80% of her free-throw attempts. If she shoots 10 free throws, what is the probability that she makes exactly 7? Use the binomial formula: $$P(X=k) = \frac{n!}{k!(n-k)!} p^k q$$^{n-k}$$.

Accepted Answer

$$P(X=7) = \frac{10!}{7!3!} (0.8)^7$ (0.2)^3$ \approx 0.201$$

Question 9

Suppose the amount of weight gained over winter is uniformly distributed from 0 to 30 lbs. What is the probability a person gains between 10 and 15 lbs?

Accepted Answer

$$P = \frac{15-10}{30-0} = \frac{5}{30} = 0.167$$

Question 10

A bank knows that 3% of all transactions are fraudulent. If 20 transactions occur in a day, what is the probability that exactly 2 are fraudulent? ($$n=20$$, $$p=0.03)$$

Accepted Answer

$$P(X=2) = \frac{20!}{2!18!} (0.03)^2$ (0.97$$)^{18}$$ \approx 0.231$$

Question 11

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

Accepted Answer

Each trial has more than two possible outcomes.

Question 12

A probability distribution is given by $$P(X=1)=0.4$$, $$P(X=2)=0.3$$, $$P(X=3)=0.3. $$What is the variance? Use $$\$$sigma^2$$ = \sum [x^2$ \cdot P(x)] - \mu^2$.

Accepted Answer

$$\mu = 1 \cdot 0.4 + 2 \cdot 0.3 + 3 \cdot 0.3 = 1.9$$; $$E[X^2$] = 1^2$ \cdot 0.4 + 2^2$ \cdot 0.3 + 3^2$ \cdot 0.3 = 4.4$$; $$\$$sigma^2$$ = 4.4 - (1.9)^2$ = 0.79$$

Question 13

A sharpshooter hits the target 4 times out of 5 on average. If he fires 5 shots, what is the probability of more than 2 hits? ($$n=5$$, $$p=0.8)$$

Accepted Answer

$$P(X\u003e2) = 1 - [P(0) + P(1) + P(2)] = 1 - [0.00032 + 0.0064 + 0.0512] = 0.942$$

Question 14

Which formula gives the mean of a binomial distribution with $$n$$ trials and probability $$p of $$success?

Accepted Answer

$$\mu = n p$$

Question 15

A random variable $$X is $$continuous if:

Accepted Answer

It can take on any value within an interval.

Question 16

Suppose you flip a coin twice. What is the probability of getting exactly one head?

Accepted Answer

$$P(X=1) = 2 \cdot (0.5)^1$ \cdot (0.5)^1$ = 0.5$$

Discrete Probability Distributions and Binomial Probability

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