,MATH 110 Module 4 Exam - Introduction to
Statistics Portage Learning - Latest (26/27)
1. A discrete probability distribution assigns probabilities to each possible outcome of a random
variable. Which condition must be satisfied for the sum of all probabilities?
A) The sum must equal the number of possible outcomes.
B) The sum must equal 1.
C) The sum must equal the expected value.
D) The sum must be less than 1.
Correct Answer: B) The sum must equal 1.
Rationale: A valid discrete probability distribution requires that the sum of all probabilities over all
possible outcomes is exactly 1, representing total certainty. The number of outcomes is a count, the
expected value is a weighted average, and a sum less than 1 would violate the fundamental rule of
probability.
2. A discrete random variable X has the probability distribution P(X=0)=0.10, P(X=1)=0.30,
P(X=2)=0.40, and P(X=3)=0.20. What is the expected value of X?
A) 1.50
B) 1.70
C) 2.00
D) 2.30
Correct Answer: B) 1.70
Rationale: The expected value is calculated as 0(0.10) + 1(0.30) + 2(0.40) + 3(0.20) = 0 + 0.30 + 0.80 +
0.60 = 1.70. Option A would be the median, C is the mode, and D results from miscalculating the
weighted sum.
3. For the probability distribution in the previous question, what is the variance of X?
,A) 0.61
B) 0.81
C) 1.00
D) 1.21
Correct Answer: B) 0.81
Rationale: Variance is E(X²) - [E(X)]². E(X²) =
0²(0.10)+1²(0.30)+2²(0.40)+3²(0.20)=0+0.30+1.60+1.80=3.70. Variance = 3.70 - (1.70)² = 3.70 - 2.89 =
0.81. Option A is the standard deviation, C is a common distractor, and D is an incorrect calculation.
4. A binomial experiment consists of 12 trials with a probability of success p = 0.25. What is the
expected number of successes?
A) 2.25
B) 3.00
C) 4.00
D) 9.00
Correct Answer: B) 3.00
Rationale: The expected value (mean) of a binomial distribution is np = 12 × 0.25 = 3.00. Option A is
np(1-p) = 2.25, which is the variance; C would be np for p=1/3; D is n(1-p) = 9.
5. The variance of a binomial random variable with n = 20 and p = 0.4 is:
A) 4.80
B) 8.00
C) 12.00
D) 2.19
Correct Answer: A) 4.80
, Rationale: The variance of a binomial distribution is np(1-p) = 20 × 0.4 × 0.6 = 4.80. Option B is np =
8.00, C is n(1-p) = 12, and D is the standard deviation (√4.80 ≈ 2.19).
6. A fair coin is flipped 6 times. What is the probability of getting exactly 4 heads?
A) 0.09375
B) 0.23438
C) 0.31250
D) 0.46875
Correct Answer: B) 0.23438
Rationale: Using the binomial formula with n=6, p=0.5, and x=4: C(6,4)(0.5)⁴(0.5)² = 15 × 0.0625 × 0.25
= 0.234375. Option A is for x=3, C is for x=2, and D is the complement of getting fewer than 4 heads.
7. In a binomial distribution, if p = 0.8 and n = 10, the distribution is:
A) Symmetric
B) Positively skewed
C) Negatively skewed
D) Uniform
Correct Answer: C) Negatively skewed
Rationale: A binomial distribution is negatively skewed when p > 0.5 (success is more likely), with a
longer tail on the left. Symmetry occurs at p = 0.5; positive skew at p < 0.5; uniform is not a
characteristic of binomial distributions.
8. Which of the following is NOT a required condition for a binomial experiment?
A) A fixed number of trials
B) Each trial has more than two possible outcomes
Statistics Portage Learning - Latest (26/27)
1. A discrete probability distribution assigns probabilities to each possible outcome of a random
variable. Which condition must be satisfied for the sum of all probabilities?
A) The sum must equal the number of possible outcomes.
B) The sum must equal 1.
C) The sum must equal the expected value.
D) The sum must be less than 1.
Correct Answer: B) The sum must equal 1.
Rationale: A valid discrete probability distribution requires that the sum of all probabilities over all
possible outcomes is exactly 1, representing total certainty. The number of outcomes is a count, the
expected value is a weighted average, and a sum less than 1 would violate the fundamental rule of
probability.
2. A discrete random variable X has the probability distribution P(X=0)=0.10, P(X=1)=0.30,
P(X=2)=0.40, and P(X=3)=0.20. What is the expected value of X?
A) 1.50
B) 1.70
C) 2.00
D) 2.30
Correct Answer: B) 1.70
Rationale: The expected value is calculated as 0(0.10) + 1(0.30) + 2(0.40) + 3(0.20) = 0 + 0.30 + 0.80 +
0.60 = 1.70. Option A would be the median, C is the mode, and D results from miscalculating the
weighted sum.
3. For the probability distribution in the previous question, what is the variance of X?
,A) 0.61
B) 0.81
C) 1.00
D) 1.21
Correct Answer: B) 0.81
Rationale: Variance is E(X²) - [E(X)]². E(X²) =
0²(0.10)+1²(0.30)+2²(0.40)+3²(0.20)=0+0.30+1.60+1.80=3.70. Variance = 3.70 - (1.70)² = 3.70 - 2.89 =
0.81. Option A is the standard deviation, C is a common distractor, and D is an incorrect calculation.
4. A binomial experiment consists of 12 trials with a probability of success p = 0.25. What is the
expected number of successes?
A) 2.25
B) 3.00
C) 4.00
D) 9.00
Correct Answer: B) 3.00
Rationale: The expected value (mean) of a binomial distribution is np = 12 × 0.25 = 3.00. Option A is
np(1-p) = 2.25, which is the variance; C would be np for p=1/3; D is n(1-p) = 9.
5. The variance of a binomial random variable with n = 20 and p = 0.4 is:
A) 4.80
B) 8.00
C) 12.00
D) 2.19
Correct Answer: A) 4.80
, Rationale: The variance of a binomial distribution is np(1-p) = 20 × 0.4 × 0.6 = 4.80. Option B is np =
8.00, C is n(1-p) = 12, and D is the standard deviation (√4.80 ≈ 2.19).
6. A fair coin is flipped 6 times. What is the probability of getting exactly 4 heads?
A) 0.09375
B) 0.23438
C) 0.31250
D) 0.46875
Correct Answer: B) 0.23438
Rationale: Using the binomial formula with n=6, p=0.5, and x=4: C(6,4)(0.5)⁴(0.5)² = 15 × 0.0625 × 0.25
= 0.234375. Option A is for x=3, C is for x=2, and D is the complement of getting fewer than 4 heads.
7. In a binomial distribution, if p = 0.8 and n = 10, the distribution is:
A) Symmetric
B) Positively skewed
C) Negatively skewed
D) Uniform
Correct Answer: C) Negatively skewed
Rationale: A binomial distribution is negatively skewed when p > 0.5 (success is more likely), with a
longer tail on the left. Symmetry occurs at p = 0.5; positive skew at p < 0.5; uniform is not a
characteristic of binomial distributions.
8. Which of the following is NOT a required condition for a binomial experiment?
A) A fixed number of trials
B) Each trial has more than two possible outcomes
