Probability and Statistics Practice Problems and Solutions

Probability Fundamentals

• Mutually Exclusive Events: If A and B are mutually exclusive, then P(A ∪ B) ≠ 0.
• Independent Events: Events are independent if their probabilities match, such as P(A) = P(B) or P(A|B) = P(A).
• Conditional Probability Example: Find P(driver is age 65+ | driver is female) = P(driver is female and 65+) / P(driver is female).
• Key Formulas:
  • P(S ∩ Y) = P(S ∩ Y) / Total
  • P(S|Y) = P(S ∩ Y) / P(Y)
  • P(Y ∪ S) = P(Y) + P(S) - P(Y ∩ S)

Probability Distributions

Expected Value Calculation

The payoff (X) for a lottery game has the following probability distribution. What is the expected value of X? (Hint: Find the value of 'a' first.)

Given values: 2 (0.4), 5 (0.3), 7 (0.2), 10 (a).
Calculation: 1 - (0.4 + 0.3 + 0.2) = 0.1, so a = 0.1.
Expected Value: 2(0.4) + 5(0.3) + 7(0.2) + 10(0.1) = 4.7

Binomial Distribution Scenarios

Which of the following scenarios can be described using a binomial distribution model for the random variable X?

• a) X = the number of phone calls received in a one-hour period.
• b) X = the number of people in a random sample of size 50 from a large population that have type-AB blood.
• c) A fair coin is flipped multiple times. Let X = the number of flips until the first tail.
• d) X = the number of speeding tickets given at a location in a year.
• e) 10 people (5 men, 5 women). Three are selected for a committee. Let X = the number of men on the committee.

Normal Distribution Characteristics

All of the following are characteristics of the normal distribution, except:

• a) Bell-shaped and symmetric about the mean.
• b) It does not take negative values (it can take all values).
• c) Total area under the curve is always one.
• d) It is a continuous distribution whose graph depends on the mean and standard deviation.
• e) The probability that x is equal to any specific value is zero.

Understanding Z-Scores

Suppose a normal distribution model describes student scores in a history class. Mary has a standardized score (z-score) of +2.5. This means that Mary’s score:

• a) Has a standard deviation of 2.5.
• b) Is 2.5 percent above average for the class.
• c) Is 2.5 standard deviations above average for the class.
• d) Is 2.5 times the average for the class.
• e) None of the above.