Probability

Probability is a mathematical framework used to quantify the likelihood of events. How probabilities are calculated depends heavily on how the events relate to one another.

Key Takeaways

• Independent events (P(A∩B) = P(A) × P(B)) don't affect each other's probability; dependent events do.

• Mutually exclusive events can't occur together, so their combined probability is a simple sum: P(A∪B) = P(A) + P(B).

• When events overlap, subtract the shared probability to avoid double-counting: P(A∪B) = P(A) + P(B) − P(A∩B).

• A set of events is exhaustive when it accounts for every possible outcome, and the total probability across all outcomes must equal 1.

Independent vs. Dependent Events

Events can be classified as independent or dependent, and this distinction determines how their probabilities are calculated.

Independent events are those where the outcome of one event does not affect the probability of the other — flipping a coin and rolling a die are independent, since the coin flip has no influence on the die roll.

For independent events, the probability of both occurring simultaneously is the product of their individual probabilities:

P(A∩B) = P(A) × P(B)

MCAT Callout — Worked Example: Independent Events: P(flipping heads) = 0.5. P(rolling a six) = 1/6. P(heads and a six) = 0.5 × (1/6) = 1/12.

Dependent events are those where the outcome of one event affects the probability of the other. Drawing a card from a deck without replacement is a classic example: each draw changes the probabilities for the next, since the total number of cards decreases.

Mutually Exclusive Events and the Addition Rule

Mutually exclusive events are events that cannot happen at the same time. When rolling a die, rolling a 2 and rolling a 5 are mutually exclusive — the die can only show one face at a time.

For mutually exclusive events, the probability of either event A or event B occurring is the sum of their individual probabilities:

P(A∪B) = P(A) + P(B)

When Events Overlap

If events are not mutually exclusive — meaning they can occur together — calculating P(A∪B) requires subtracting the overlap to avoid double-counting. This is the addition rule for overlapping events:

P(A∪B) = P(A) + P(B) − P(A∩B)

MCAT Callout — Worked Example: The Addition Rule for Overlapping Events: 20% of students play basketball: P(A) = 0.2. 30% play soccer: P(B) = 0.3. 10% play both: P(A∩B) = 0.1. P(plays basketball, soccer, or both) = 0.2 + 0.3 − 0.1 = 0.4 (40% of the class).

Exhaustiveness

A set of events is exhaustive when it covers all possible outcomes in the sample space. Rolling a die, the events "rolling an even number" and "rolling an odd number" are exhaustive, since together they account for all six outcomes. Exhaustiveness matters because it guarantees the total probability of all possible outcomes equals 1 — a fundamental rule of probability.

Common MCAT Mistakes

• Adding probabilities for independent events instead of multiplying. P(A∩B) = P(A) × P(B) only for the joint probability of two independent events — adding them (a mutually-exclusive-event rule) gives the wrong answer.

• Forgetting to subtract the overlap for non-mutually-exclusive events. P(A∪B) = P(A) + P(B) − P(A∩B) whenever the events can happen together; skipping the subtraction double-counts the overlap.

• Treating dependent events like independent ones. Drawing cards without replacement changes the probabilities for each subsequent draw — the individual-probability product only works when outcomes truly don't affect each other.

• Confusing mutually exclusive with independent. Mutually exclusive events can't happen together at all (rolling a 2 and a 5 on one die roll); independent events can both happen, just without influencing each other's probability (a coin flip and a die roll).

MCAT-Style Concept Check

Question: A bag contains only red and blue marbles. A student draws one marble, does not replace it, then draws a second marble. Which statement correctly describes these two draws?

• A) The draws are independent because each draw has the same possible outcomes.

• B) The draws are dependent because the first draw changes the probabilities for the second.

• C) The draws are mutually exclusive because only one marble can be drawn at a time.

• D) The draws are exhaustive because every marble in the bag could be drawn.

Answer: B

Explanation: Removing a marble without replacement changes the total number of marbles and the ratio of red to blue remaining, so the probability of the second draw's outcome depends on the first draw's result — the definition of dependent events. The draws aren't independent (A), since the first draw does affect the second's probabilities. "Mutually exclusive" (C) describes events that can't both happen in a single trial, not two sequential draws. "Exhaustive" (D) describes whether a set of outcomes covers every possibility, not a relationship between two draws.

FAQ

What's the difference between independent and dependent events?

Independent events don't affect each other's probability — the outcome of one has no bearing on the other, like flipping a coin and rolling a die. Dependent events do affect each other — like drawing cards from a deck without replacement, where each draw changes the odds for the next.

How do you calculate the probability of two independent events both happening?

Multiply their individual probabilities: P(A∩B) = P(A) × P(B).

When do you need to subtract P(A∩B) in the addition rule?

Only when the events are not mutually exclusive — meaning they can occur together. If the events can overlap, P(A∪B) = P(A) + P(B) − P(A∩B); if they're mutually exclusive, the simple sum P(A∪B) = P(A) + P(B) is correct.

What does it mean for a set of events to be exhaustive?

A set of events is exhaustive when it covers every possible outcome in the sample space, which means the probabilities of all those outcomes must add up to 1.