📝 Chapter 4: Probability — Theoretical & Empirical

Assessment AS Learning — Practice Quiz

🔄 Not Graded — Unlimited Retakes

Purpose: Self-check probability fundamentals: theoretical/empirical/subjective, conditional probability, expected value, variance.

Score: 0 / 12

Topic 4.1 — Theoretical & Empirical Probability

Question 1

A coin is flipped 1000 times and lands on heads 540 times. What type of probability is $ \dfrac{540}{1000} = 0.54 $?

Theoretical
Empirical (relative frequency)
Subjective
Conditional

Solution:

This is empirical probability — based on observed data from repeated experiments. The theoretical probability is 0.5.

Question 2

Two fair dice are rolled. What is the theoretical probability that the sum is 7?

Solution:

Sample space: 36 outcomes. Sum = 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) — 6 outcomes. $ P = \dfrac{6}{36} = \dfrac{1}{6} \approx 0.167 $.

Topic 4.2 — Conditional Probability

Question 3

If $ P(A) = 0.5 $, $ P(B) = 0.4 $, $ P(A \cap B) = 0.2 $, then $ P(B \mid A) = ? $

Solution:

$ P(B \mid A) = \dfrac{P(A \cap B)}{P(A)} = \dfrac{0.2}{0.5} = 0.4 $.

Question 4

A card is drawn from a deck. Given that it is a face card, what is the probability it is a King?

$ \tfrac{1}{13} $
$ \tfrac{1}{3} $
$ \tfrac{1}{4} $
$ \tfrac{4}{52} $

Solution:

Given face card (12 face cards), 4 are Kings. $ P(K \mid F) = \dfrac{4}{12} = \dfrac{1}{3} $.

Topic 4.3 — Discrete Probability Distributions

Question 5

A random variable $ X $ has the distribution:
$ X = 0 $: $ P = 0.2 $ $ X = 1 $: $ P = 0.3 $ $ X = 2 $: $ P = 0.4 $ $ X = 3 $: $ P = 0.1 $
Verify it is valid by checking the sum. Then find $ P(X \ge 2) $.

Solution:

Sum: $ 0.2+0.3+0.4+0.1 = 1.0 $ ✓. $ P(X \ge 2) = P(2) + P(3) = 0.4 + 0.1 = 0.5 $.

Topic 4.4 — Expected Value

Question 6

A game costs $5 to play. You roll a single die. If you roll a 6, you win $20. Otherwise, you win nothing. Find the expected NET value (winnings minus cost) per play.

Solution:

$ E(\text{net}) = \dfrac{1}{6}(20 - 5) + \dfrac{5}{6}(0 - 5) = \dfrac{15}{6} - \dfrac{25}{6} = -\dfrac{10}{6} \approx -\$1.67 $. The game is unfair (loses money on average).

Question 7

A random variable $ X $ takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find $ E(X) $.

Solution:

$ E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1.0 + 0.9 = 2.1 $.

Topic 4.5 — Variance & Standard Deviation

Question 8

For $ X $ in Q7, find $ E(X^2) $.

Solution:

$ E(X^2) = 1^2(0.2) + 2^2(0.5) + 3^2(0.3) = 0.2 + 2.0 + 2.7 = 4.9 $.

Question 9

For $ X $ in Q7, find $ \mathrm{Var}(X) = E(X^2) - [E(X)]^2 $.

Solution:

$ \mathrm{Var}(X) = 4.9 - (2.1)^2 = 4.9 - 4.41 = 0.49 $. So $ \sigma = \sqrt{0.49} = 0.7 $.

Mixed

Question 10

A bag has 3 red and 5 blue marbles. Two are drawn without replacement. What is $ P(\text{both red}) $? (decimal to 4 places)

Solution:

$ P = \dfrac{3}{8} \cdot \dfrac{2}{7} = \dfrac{6}{56} = \dfrac{3}{28} \approx 0.1071 $.

Question 11

An insurance company pays $10 000 if a customer dies during the year. The probability of death is 0.001. To make the policy fair (zero expected profit), what should the premium be?

Solution:

Expected payout = $ 0.001 \times 10\,000 = \$10 $. Fair premium = $10. (Real premiums are higher to cover overhead and profit.)

Question 12

In a 30-question MC test (4 choices each), a student guesses on every question. The expected number correct is:

Solution:

$ E = np = 30 \cdot \tfrac{1}{4} = 7.5 $.