📝 Chapter 1: Counting & Sample Spaces

A coin is tossed 3 times. How many outcomes are in the sample space?

Two events \( A \) and \( B \) are mutually exclusive if:

A meal is built by choosing 1 of 3 appetizers, 1 of 5 mains, and 1 of 4 desserts. How many possible meals?

A card is drawn from a 52-card deck. \( P(\text{Spade or Ace}) = ? \)

In a school, 60% take Math, 40% take Science, 25% take both. The percentage taking neither is:

Two cards are drawn without replacement from a 52-card deck. \( P(\text{both Hearts}) = ? \)

A 3-letter code uses letters from \( \{A,B,C,D,E\} \) without repetition. How many codes are possible?

In a class of 100 students, 50 play hockey, 60 play soccer, and 30 play both. Use a Venn diagram to determine how many play exactly one of the two sports.

A bag contains 6 red, 4 blue, and 2 green marbles. Two marbles are drawn without replacement. What is the probability that exactly one is red? Show your work using a tree diagram or case analysis.

A 4-digit code uses digits 0–9. How many codes are even (last digit is 0, 2, 4, 6, or 8) AND have all distinct digits?

If \( P(A) = 0.4 \), \( P(B) = 0.5 \), and \( P(A \cap B) = 0.2 \), determine if \( A \) and \( B \) are independent. Justify your answer using the test for independence.

A student says: "If \( P(A) = 0.6 \) and \( P(B) = 0.5 \), then \( P(A \cup B) = 1.1 \), but probabilities can't exceed 1, so this is impossible." What is wrong with the student's reasoning? Provide a correct method.

Explain the difference between mutually exclusive events and independent events. Give an example of each. Can two events be both mutually exclusive AND independent? Justify.

Describe a step-by-step procedure for solving a "without replacement" probability problem. Include: identifying dependence, the formula, and the role of conditional probability.

Compare a tree diagram and a Venn diagram. When is each more useful? Give a small example for each.

An online retailer's password requires 6 characters: the first must be a letter (A–Z, case-insensitive), the rest may be letters or digits, with repetition allowed. How many passwords are possible?

In a survey of 200 commuters: 120 take the bus, 90 take the subway, 50 take both. a) How many take the bus only? b) How many take neither? c) If a commuter is selected at random, find \( P(\text{bus only}) \).

In a quality-control test, 5% of widgets are defective. A widget is selected and tested. The test correctly identifies a defective widget 90% of the time and a good widget 95% of the time. Use a tree diagram to find \( P(\text{test reports defective}) \).

A medical screening test has \( P(\text{positive} \mid \text{disease}) = 0.99 \) and \( P(\text{positive} \mid \text{no disease}) = 0.05 \). The disease prevalence is 1%. If a randomly chosen person tests positive, what is \( P(\text{disease} \mid \text{positive}) \)? Use a tree diagram.