3.3 Probability Rules, Odds, and Expected Value

From counts to probabilities

A probability is a part-whole comparison: P(event) = favourable outcomes / total outcomes. The hardest part is often not the fraction; it is defining the event and the sample space. Math 30-2 questions may give a Venn diagram, a two-way table, a tree diagram, a counting situation, or a verbal description. In each form, first identify the total outcomes that are possible under the question's conditions. Then identify the outcomes that satisfy the event.

The formula sheet includes probability rules for unions and intersections, but you must choose the rule from the wording. A or B points to a union. A and B points to an intersection. Not A points to the complement. Given A points to conditional probability, which is the next section.

Mutually exclusive versus overlapping

Two events are mutually exclusive when they cannot happen on the same trial. Drawing one card and getting a heart or a club is mutually exclusive because one card cannot be both suits. Drawing one card and getting a heart or a face card is not mutually exclusive because a jack of hearts fits both descriptions.

Worked example: In a group of 80 students, 34 play volleyball, 29 play basketball, and 12 play both. What is the probability a randomly selected student plays volleyball or basketball?

Use the union rule with overlap: 34 + 29 - 12 = 51 students play at least one of the two sports. The probability is 51/80 = 0.6375, or 63.75 percent.

If you used 34 + 29 = 63, the 12 students who play both were counted twice. If you subtracted from 80 first, you were probably finding neither sport rather than at least one sport.

Odds are part-part

Odds in favour of an event compare favourable outcomes to unfavourable outcomes. Probability compares favourable outcomes to total outcomes. This distinction matters because the total is not the second number in an odds statement; the total is the sum of the two parts.

If the odds in favour of winning a draw are 5:11, then the probability of winning is 5/(5+11) = 5/16. The probability of not winning is 11/16. The odds against winning are 11:5.

To convert probability to odds, split the total into favourable and unfavourable parts. If P(A) = 3/8, then 3 parts are favourable and 5 parts are unfavourable, so the odds in favour are 3:5. Alberta's bulletin commentary specifically notes that students can struggle more with odds statements than probability statements, so slow down when you see a colon.

Expected value

Expected value is the long-run average outcome per trial. Multiply each outcome value by its probability, then add the products. In a game, use net gain or net loss if the cost to play is included.

Worked example: A fundraiser spinner has four equal sections. A player pays $2 to play. The prizes are $0, $0, $3, and $9. What is the expected net gain for the player?

Net outcomes after paying $2 are -2, -2, 1, and 7. Each has probability 1/4. Expected value = (-2)(1/4) + (-2)(1/4) + (1)(1/4) + (7)(1/4) = 1. The player averages a $1 gain per play over many plays. The fundraiser averages a $1 loss per play if the model is accurate.

Expected value is not the most likely outcome. In this example, the most common net result is a $2 loss, but the large prize raises the long-run average.

Combining counting and probability

Some questions require a count before the probability rule. Suppose a four-digit code uses digits 1 through 6 with no repetition. What is the probability the code begins with an even digit? Total codes: 6P4 = 6 times 5 times 4 times 3 = 360. Favourable codes: choose the first digit from 2, 4, 6, giving 3 choices, then fill the next three slots with 5, 4, and 3 choices. Favourable count: 180. Probability: 180/360 = 1/2.

Diploma traps

Trap 1: treating every or as simple addition. Simple addition works only when events cannot overlap. If overlap is possible, subtract it once.

Trap 2: reading odds as probability. Odds of 2:3 do not mean probability 2/3. The total is 2 + 3 = 5, so the probability is 2/5.

Trap 3: using percent values inconsistently. Convert all probabilities to fractions, decimals, or percentages before combining them. Mixing formats creates arithmetic errors.

Trap 4: ignoring the sample space after a restriction. If the question says among students who play a sport, the total is not the whole school. That is conditional probability, not a basic union problem.

A good scratch-work line names the event, the total, and the rule: volleyball or basketball, overlap possible, use union with subtraction. That one sentence usually points to the correct calculation.