📝 Chapter 4: Probability — Theoretical & Empirical

A coin is tossed 4 times. \( P(\text{exactly 2 heads}) \) = ? (Decimal to 4 places)

If \( P(A) = 0.3 \), \( P(B) = 0.4 \), and \( A, B \) are independent, then \( P(A \cap B) \) = ?

A discrete distribution: \( P(X=0)=0.1, P(X=1)=0.3, P(X=2)=0.4, P(X=3)=0.2 \). Find \( E(X) \).

For the distribution in Q3, find \( \mathrm{Var}(X) \).

A 6-sided die is rolled. \( P(\text{rolling } 5 \mid \text{rolling odd}) = ? \)

A class has 60% pass rate. If 5 students are selected, \( P(\text{exactly 4 pass}) \) = ? (decimal to 4 places). Hint: this is \( \binom{5}{4}(0.6)^4(0.4)^1 \).

A fair 10-sided die is rolled. The expected value of one roll is:

In a casino game, you bet $5. With probability 0.45 you win $10 (net +$5); with probability 0.55 you lose your bet (net -$5). Compute the expected value per play. After 100 plays, what is the expected loss?

A box contains 10 light bulbs, of which 3 are defective. Two bulbs are drawn without replacement. Construct the probability distribution for \( X \) = number of defective bulbs drawn. Compute \( E(X) \) and verify that \( E(X) = n \cdot \dfrac{r}{N} \) (the hypergeometric mean).

In a class, 70% of students did the homework. Of those who did the homework, 90% passed the test. Of those who didn't, only 30% passed. If a randomly chosen student passed, find the probability they did the homework. Use a tree diagram and Bayes' rule (or table).

Two events have \( P(A) = 0.6 \), \( P(B) = 0.4 \), \( P(A \cup B) = 0.7 \). Determine \( P(A \cap B) \), then test whether \( A \) and \( B \) are independent.

A student asserts: "If \( P(A \mid B) = P(A) \), then \( P(B \mid A) = P(B) \)." Is this true? Justify with the multiplication rule and explain in plain language what independence means.

Explain in plain English what "expected value" means in the long run, using a small example (e.g., a coin-flipping game). Why is it not necessarily the most likely outcome?

Compare theoretical, empirical, and subjective probability. Give an example of each, and describe the relationship between theoretical probability and empirical probability as the number of trials grows.

Describe the conditions under which a fair game has \( E = 0 \). Give an example of an unfair game (\( E < 0 \) for the player) that is commonly played.

A car insurance company charges $800 per year. The probability of a $20 000 claim during the year is 0.02; the probability of a $5 000 claim is 0.05; otherwise no claim. What is the expected profit per policyholder?

A medical screening test for a disease has sensitivity 0.95 and specificity 0.90. The disease prevalence is 2%. If a person tests positive, what is the probability they have the disease? (Bayes' theorem.)

In a roulette game (American wheel, 38 slots), a player bets $1 on red. There are 18 red, 18 black, and 2 green slots. If red comes up, the player wins $1 (net +$1); otherwise loses $1. Find the expected value and explain why casinos profit.

A weather forecaster says there is a 70% chance of rain tomorrow. The cost of carrying an umbrella (when it doesn't rain) is $2 (annoyance), and the cost of not carrying one (when it rains) is $20. Should you carry the umbrella? Use expected value to compare the two strategies.