5.3 Probability

The Language of Chance

Probability measures how likely an event is on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, a decimal, or a percent.

Basic Probability

For equally likely outcomes,

P(event) = (number of favorable outcomes) / (total number of outcomes).

Worked example: A standard die has 6 faces. P(rolling a 5) = 1/6 ≈ 0.167. P(rolling an even number) = 3/6 = 1/2 = 0.5, because {2, 4, 6} are the favorable outcomes.

Worked example: A bag holds 4 red, 3 blue, and 5 green marbles — 12 total. P(blue) = 3/12 = 1/4 = 0.25.

Complements

The complement of an event is everything that is not the event. Because something must happen,

P(not A) = 1 - P(A).

Worked example: If P(rain) = 0.30, then P(no rain) = 1 - 0.30 = 0.70. For the marble bag, P(not blue) = 1 - 3/12 = 9/12 = 3/4 = 0.75. The complement is a shortcut whenever the words "at least one" or "not" appear.

Mutually Exclusive Events (Addition)

Two events are mutually exclusive if they cannot happen at the same time (rolling a 2 and rolling a 5 on one roll). For mutually exclusive events,

P(A or B) = P(A) + P(B).

Worked example: On one die, P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3.

If the events can overlap, subtract the overlap: P(A or B) = P(A) + P(B) - P(A and B). Drawing a card that is a king or a heart gives P = 4/52 + 13/52 - 1/52 = 16/52 = 4/13, subtracting the king of hearts that was counted twice.

Independent vs. Dependent Events (Multiplication)

Events are independent if one does not affect the other (separate coin flips). For independent events,

P(A and B) = P(A) × P(B).

Worked example: Flip a coin and roll a die. P(heads and a 4) = 1/2 × 1/6 = 1/12.

Worked example (independent, "with replacement"): Draw a marble from the 12-marble bag, put it back, then draw again. P(red then red) = 4/12 × 4/12 = 1/3 × 1/3 = 1/9.

Events are dependent when the first outcome changes the second, and drawing without replacement is the classic case.

Worked example: From 4 red marbles out of 12, draw two without replacing the first. P(red then red) = 4/12 × 3/11 = (4 × 3) / (12 × 11) = 12/132 = 1/11. The second fraction became 3/11 because one red marble, and one marble overall, were removed.

Putting the Rules Together

Many questions combine rules. To find the probability of at least one success, it is usually easier to find the complement — the probability of zero successes — and subtract from 1.

Worked example: You flip a fair coin 3 times. What is P(at least one heads)? Rather than list cases, use the complement. Because the flips are independent, P(no heads) = P(tails) × P(tails) × P(tails) = 1/2 × 1/2 × 1/2 = 1/8. Then P(at least one heads) = 1 - 1/8 = 7/8.

Worked example (spinner): A spinner has 8 equal sections numbered 1 to 8. Since the sections are equally likely, P(a number greater than 5) = P(6, 7, or 8). These three outcomes are mutually exclusive, so add: 1/8 + 1/8 + 1/8 = 3/8. By the complement rule, P(5 or less) = 1 - 3/8 = 5/8. One spinner shows the addition rule and the complement rule working together.

Expected Value

Expected value is the long-run average outcome. Multiply each outcome by its probability, then add the products:

Expected value = Σ (outcome × probability).

Worked example: A game pays $10 with probability 0.2, $2 with probability 0.5, and $0 with probability 0.3. Expected value = 10(0.2) + 2(0.5) + 0(0.3) = 2 + 1 + 0 = $3. Each play averages $3 over the long run. If a ticket costs $4, the game loses 4 - 3 = $1 per play on average, so it is not worth playing.

Worked example (fair die): Expected value of one roll = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21/6 = 3.5. The average is 3.5 even though you can never actually roll a 3.5.

Quick Checks

• Every probability lands between 0 and 1; if you ever get a value like 1.5, stop and recheck your work.
• "And" usually means multiply the probabilities; "or" usually means add them.
• Use the complement whenever the phrase "at least one" appears in the question.