Probability Study Guide

Probability is the branch of math that measures how likely an event is to happen. A probability study guide helps you connect everyday ideas like chance, risk, and randomness to formal tools such as sample spaces, events, complements, conditional probability, and expected value. Use this guide to review core probability notes, practice common formulas, and test your understanding with flashcards and a probability quiz.

Key takeaways

• Probability values range from 0 to 1, where 0 means impossible and 1 means certain.
• The probability of an event is often calculated as favorable outcomes divided by total equally likely outcomes.
• Key probability rules include the complement rule, addition rule, multiplication rule, and conditional probability formula.
• Independent events do not affect each other, while dependent events do affect each other’s probabilities.
• Expected value describes the long-run average outcome of a random process.

What Probability Measures

Probability measures the likelihood that an event will occur. An event is any outcome or set of outcomes from a random process, such as rolling a die, flipping a coin, or drawing a card. The sample space is the complete set of possible outcomes. For example, when rolling a standard six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. The probability of rolling a 4 is 1/6 because there is one favorable outcome out of six equally likely outcomes.

Basic Probability Formula and Complement Rule

The basic probability formula is P(event) = number of favorable outcomes / total number of equally likely outcomes. For example, the probability of drawing a heart from a standard deck of 52 cards is 13/52, or 1/4. The complement of an event is everything outside that event. If P(A) is the probability that event A happens, then P(not A) = 1 – P(A). If the probability of rain is 0.30, the probability that it does not rain is 0.70.

Addition Rule, Multiplication Rule, and Conditional Probability

The addition rule is used when finding the probability that one event or another event occurs. For any events A and B, P(A or B) = P(A) + P(B) – P(A and B). If events are mutually exclusive, they cannot happen at the same time, so P(A and B) = 0. The multiplication rule helps find the probability that two events both occur. For independent events, P(A and B) = P(A) × P(B). Conditional probability measures the chance of an event given that another event has already happened: P(A given B) = P(A and B) / P(B), as long as P(B) is not 0.

Independent vs. Dependent Events

Two events are independent if the outcome of one event does not change the probability of the other. For example, flipping a coin and rolling a die are independent because the coin result does not affect the die result. Events are dependent if one event changes the probability of another. Drawing two cards from a deck without replacement is dependent because the first card changes what remains in the deck for the second draw.

Expected Value and Probability Distributions

Expected value is the weighted average of possible outcomes, where each outcome is weighted by its probability. It is useful for games, decisions, and long-run predictions. To calculate expected value, multiply each outcome by its probability and add the products: E(X) = Σ[x × P(x)]. For example, if you win $10 with probability 0.20 and $0 with probability 0.80, the expected value is 10(0.20) + 0(0.80) = $2. This does not mean you win $2 every time; it means the average outcome over many trials is $2.

Flashcards

Quiz

1. What is the probability of flipping heads on a fair coin?

1. A. 0
2. B. 1/4
3. C. 1/2
4. D. 1

2. When rolling a standard six-sided die, what is the probability of rolling an even number?

1. A. 1/6
2. B. 1/3
3. C. 1/2
4. D. 2/3

3. If P(A) = 0.35, what is P(not A)?

1. A. 0.35
2. B. 0.50
3. C. 0.65
4. D. 1.35

4. Which pair of events is most likely independent?

1. A. Drawing two cards from a deck without replacement
2. B. Flipping a coin and rolling a die
3. C. Choosing two students from a class without replacement
4. D. Taking one marble and then another from a bag without replacement

5. What is the formula for conditional probability P(A given B)?

1. A. P(A) + P(B)
2. B. P(A and B) / P(B)
3. C. P(A) × P(B)
4. D. 1 – P(A)

6. If you win $5 with probability 0.40 and $0 with probability 0.60, what is the expected value?

1. A. $0.60
2. B. $2.00
3. C. $5.00
4. D. $6.00

FAQs

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