Q: A gambler bets $$5 on a roulette outcome with a probability of $$\frac{5}{38}$$ to win 30$$, and a probability of $$\frac{33}{38} to $$lose $5. What is the expected value of this bet?

$$E = 30 \times \frac{5}{38} + (-5) \times \frac{33}{38}$$

Q: On a multiple-choice test with 17 questions and 4 answer choices per question, what is the expected number of correct answers for a student who guesses on all questions?

$$E = 17 \times \frac{1}{4} = 4.25$$

Q: If $$X is a $$binomial random variable with $$n = 17$$ and $$p = 0.25$$, what is the variance of $$X$$?

$$\$$sigma^2$$ = n p (1-p) = 17 \times 0.25 \times 0.75$$

Q: 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 $$E(X)$$?

$$E(X) = 0 \times 0.1 + 1 \times 0.3 + 2 \times 0.4 + 3 \times 0.2$$

Question 1

Which of the following are requirements for a table to represent a probability distribution?

Accepted Answer

The sum of all probabilities must equal 1.

Question 2

Given the probability distribution:

x: 0, 1, 2, 3
P(x): 0.351, 0.438, 0.190, 0.021

Does this table represent a valid probability distribution?

Accepted Answer

Yes, the table shows a probability distribution.

Question 3

A gambler bets $$5 on a roulette outcome with a probability of $$\frac{5}{38}$$ to win 30$$, and a probability of $$\frac{33}{38} to $$lose $5. What is the expected value of this bet?

Accepted Answer

$$E = 30 	imes \frac{5}{38} + (-5) 	imes \frac{33}{38}$$

Question 4

If $$p = 0.64 is $$the probability that an adult uses a smartphone in meetings or classes, what is the probability that exactly 3 out of 7 randomly selected adults use their smartphones in meetings or classes?

Accepted Answer

$$P(X=3) = \binom{7}{3} (0.64)^3$ (0.36)^4$

Question 5

On a multiple-choice test with 17 questions and 4 answer choices per question, what is the expected number of correct answers for a student who guesses on all questions?

Accepted Answer

$$E = 17 	imes \frac{1}{4} = 4.25$$

Question 6

A bone density test score is normally distributed with mean $$\mu = 0$$ and $$\sigma = 1. $$What is the probability that a randomly selected score is greater than $$0.69$$?

Accepted Answer

$$P(Z \u003e 0.69) = 1 - P(Z \u003c 0.69)$$

Question 7

For a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$, what is the $$z-$$score for an IQ score of $$122$$?

Accepted Answer

$$z = \frac{122 - 100}{15}$$

Question 8

Which of the following is a biased estimator?

Accepted Answer

Median

Question 9

If the original population is normally distributed, what can be said about the distribution of sample means for any sample size?

Accepted Answer

The distribution of sample means is normal for any sample size.

Question 10

A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of $$200$$ and a standard deviation of $$50. $$What is the probability that a randomly selected applicant has a rating between $$200$$ and $$275$$?

Accepted Answer

$$P(200 \u003c X \u003c 275) = P\left(0 \u003c Z \u003c \frac{275-200}{50}\right)$$

Question 11

If $$X is a $$binomial random variable with $$n = 17$$ and $$p = 0.25$$, what is the variance of $$X$$?

Accepted Answer

$$\$$sigma^2$$ = n p (1-p) = 17 	imes 0.25 	imes 0.75$$

Question 12

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

$$E(X) = 0 	imes 0.1 + 1 	imes 0.3 + 2 	imes 0.4 + 3 	imes 0.2$$

Question 13

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

Accepted Answer

The random variable must be categorical.

Question 14

If the mean pulse rate for females is $$\mu = 73.0$$ and $$\sigma = 12.5$$, what is the probability that a randomly selected female has a pulse rate less than $$80$$?

Accepted Answer

$$P(X \u003c 80) = P\left(Z \u003c \frac{80-73.0}{12.5}\right)$$

Question 15

Which of the following statements about the Central Limit Theorem (CLT) is correct?

Accepted Answer

The CLT states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 16

If the overhead reach distances of adult females are normally distributed with $$\mu = 200 cm $$and $$\sigma = 7.8 cm$$, what is the $$z-$$score for a reach distance of $$213.9 cm$$?

Accepted Answer

$$z = \frac{213.9 - 200}{7.8}$$

Question 17

Which of the following is an unbiased estimator?

Accepted Answer

Sample mean

Question 18

If $$X is a $$binomial random variable with $$n = 10$$ and $$p = 0.3$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \sqrt{n p (1-p)} = \sqrt{10 	imes 0.3 	imes 0.7}$$

Probability Distributions, Expected Value, Binomial, and Normal Distribution Study Guide

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