MATH 110 MODULE 4 - PROBABILITY
DISTRIBUTIONS AND THE NORMAL
CURVE 2026/2027 UPDATE. JUST
RELEASED
1. Which of the following is a fundamental requirement for a discrete probability
distribution?
A. The variance must equal the mean of the distribution.
B. Each individual probability must be greater than 0.5.
C. The random variable must be able to take on any value within an interval.
D. The sum of all probabilities must be exactly 1.0.
Answer: D
Conceptual Explanation: For a discrete probability distribution to be valid, two conditions
must be met: each probability P(x) must be between 0 and 1 inclusive, and the sum of all
probabilities in the sample space must equal 1.
2. A discrete random variable X has a probability distribution where P(X=1)=0.2, P(X=2)=0.5,
and P(X=3)=0.3. What is the mean (expected value) of X?
A. 2.1
B. 2.0
,C. 2.5
D. 1.8
Answer: A
Conceptual Explanation: The mean is calculated as E(X) = Σ [x * P(x)]. So, (10.2) + (20.5) +
(3*0.3) = 0.2 + 1.0 + 0.9 = 2.1.
3. In a binomial experiment with n = 10 and p = 0.4, what is the probability of exactly 3
successes?
A. 0.2150
B. 0.2007
C. 0.2508
D. 0.1209
Answer: A
Conceptual Explanation: Using the binomial formula P(x) = nCx * p^x * q^(n-x): 10C3 *
(0.4)^3 * (0.6)^7 = 120 * 0.064 * 0.02799 = 0.21499… rounds to 0.2150.
4. Which property distinguishes a Poisson distribution from a Binomial distribution?
A. The trials are dependent on each other.
B. There are a fixed number of trials n.
C. The probability of success changes with every trial.
, D. It measures the number of occurrences over a specific interval of time or space.
Answer: D
Conceptual Explanation: The Poisson distribution models the number of events occurring
in a fixed interval of time or space, whereas the Binomial distribution models the number
of successes in a fixed number of independent trials.
5. If X follows a Poisson distribution with a mean of 4, what is the standard deviation of X?
A. 4.0
B. 16.0
C. 2.0
D. 1.0
Answer: C
Conceptual Explanation: For a Poisson distribution, the variance is equal to the mean (λ).
Since the mean is 4, the variance is 4, and the standard deviation is the square root of 4,
which is 2.
6. A continuous random variable X follows a normal distribution. What is the probability that
X is exactly equal to its mean?
A. 0.50
B. 1.00
C. 0.00
DISTRIBUTIONS AND THE NORMAL
CURVE 2026/2027 UPDATE. JUST
RELEASED
1. Which of the following is a fundamental requirement for a discrete probability
distribution?
A. The variance must equal the mean of the distribution.
B. Each individual probability must be greater than 0.5.
C. The random variable must be able to take on any value within an interval.
D. The sum of all probabilities must be exactly 1.0.
Answer: D
Conceptual Explanation: For a discrete probability distribution to be valid, two conditions
must be met: each probability P(x) must be between 0 and 1 inclusive, and the sum of all
probabilities in the sample space must equal 1.
2. A discrete random variable X has a probability distribution where P(X=1)=0.2, P(X=2)=0.5,
and P(X=3)=0.3. What is the mean (expected value) of X?
A. 2.1
B. 2.0
,C. 2.5
D. 1.8
Answer: A
Conceptual Explanation: The mean is calculated as E(X) = Σ [x * P(x)]. So, (10.2) + (20.5) +
(3*0.3) = 0.2 + 1.0 + 0.9 = 2.1.
3. In a binomial experiment with n = 10 and p = 0.4, what is the probability of exactly 3
successes?
A. 0.2150
B. 0.2007
C. 0.2508
D. 0.1209
Answer: A
Conceptual Explanation: Using the binomial formula P(x) = nCx * p^x * q^(n-x): 10C3 *
(0.4)^3 * (0.6)^7 = 120 * 0.064 * 0.02799 = 0.21499… rounds to 0.2150.
4. Which property distinguishes a Poisson distribution from a Binomial distribution?
A. The trials are dependent on each other.
B. There are a fixed number of trials n.
C. The probability of success changes with every trial.
, D. It measures the number of occurrences over a specific interval of time or space.
Answer: D
Conceptual Explanation: The Poisson distribution models the number of events occurring
in a fixed interval of time or space, whereas the Binomial distribution models the number
of successes in a fixed number of independent trials.
5. If X follows a Poisson distribution with a mean of 4, what is the standard deviation of X?
A. 4.0
B. 16.0
C. 2.0
D. 1.0
Answer: C
Conceptual Explanation: For a Poisson distribution, the variance is equal to the mean (λ).
Since the mean is 4, the variance is 4, and the standard deviation is the square root of 4,
which is 2.
6. A continuous random variable X follows a normal distribution. What is the probability that
X is exactly equal to its mean?
A. 0.50
B. 1.00
C. 0.00
