5.2 Probability, Counting, and Models

Probability is organized counting

ACT Math probability questions are usually short, but they punish vague counting. The official Statistics & Probability description includes calculating probabilities by recognizing related sample spaces. A sample space is the set of outcomes you are considering. The basic formula is probability = favorable outcomes / total outcomes, and the denominator must match the situation after any condition has been applied.

Before using a formula, name the event in plain language. Is the question asking for exactly one, at least one, none, both, either, or given that something already happened? Those phrases point to different models.

Probability rules worth knowing

The complement is the highest-yield shortcut. If a basketball player has a 20% chance to make each of two independent long shots, the probability of at least one make is easier as 1 - P(no makes) = 1 - (0.80)(0.80) = 0.36. Directly listing first only, second only, and both works, but it takes longer and creates more room for omission.

Independent, dependent, and conditional

Independent events keep the same probability from one trial to the next. Rolling a number cube twice is independent because the first roll does not change the second roll. Drawing two cards without replacement is dependent because the first draw changes both the numerator and denominator for the second draw.

Conditional probability changes the denominator by design. Suppose a table shows 18 juniors in band, 12 seniors in band, 6 juniors not in band, and 14 seniors not in band. If the selected student is known to be a senior, ignore juniors entirely. The senior sample space is 12 + 14 = 26 students, and the probability that the senior is in band is 12/26 = 6/13.

The ACT trap is using the whole table total when the phrase among seniors, given senior, or of the seniors already narrowed the group.

Counting: multiplication, permutations, combinations

The Fundamental Counting Principle says that if one choice can happen in a ways and another independent choice can happen in b ways, the combined number of outcomes is ab. A 4-shirt, 3-pants, 2-shoes outfit count is 4 x 3 x 2 = 24 outfits. This is not a probability yet; it is the denominator or the favorable count, depending on the question.

Use a permutation when order matters. Arranging 3 runners from 8 into first, second, and third place gives 8 x 7 x 6 = 336 ordered finishes. Use a combination when order does not matter. Choosing 3 students from 8 for a committee gives 8C3 = 56 groups because the same three people are not different just because you list them in another order.

Restrictions usually require casework. If a 4-digit code must use digits 0 through 9, cannot repeat, and cannot begin with 0, the first digit has 9 choices, then the remaining places have 9, 8, and 7 choices. The count is 9 x 9 x 8 x 7, not 10 x 9 x 8 x 7, because the first position has a special restriction.

Probability models and expected reasonableness

A probability model assigns probabilities to outcomes. Every probability must be between 0 and 1, and all probabilities in the complete model must sum to 1. If a spinner model lists red 0.25, blue 0.30, green 0.20, and yellow 0.35, the model is impossible because the probabilities sum to 1.10.

For ACT purposes, expected value usually means long-run average: multiply each outcome by its probability and add. If a game pays 5 points with probability 0.30 and 0 points otherwise, the expected points are 5(0.30) + 0(0.70) = 1.5. That does not mean a single play awards 1.5 points; it means the average over many plays approaches 1.5.

Common ACT probability traps

• Reusing the original denominator after drawing without replacement.
• Treating at least one as exactly one.
• Multiplying for or when the cases should be added.
• Using permutations for committees or combinations for rankings.
• Forgetting that probabilities in a full model must add to 1.

A good probability setup can be written in one line: event phrase, sample space, favorable count, rule. If any of those four pieces is unclear, do not calculate yet.

Casework without double-counting

Some counting problems cannot be done in one multiplication line because there are separate cases. The safest approach is to make cases that do not overlap, count each case, and add. For example, if a two-digit number must be made from digits 1 through 5 with no repeats and must be even, the units digit is either 2 or 4. After choosing the even units digit, the tens digit has 4 remaining choices. The count is 2 x 4 = 8.

If the cases overlap, adding directly double-counts. For a class with 14 students in art, 10 in music, and 4 in both, the number in art or music is 14 + 10 - 4 = 20. The overlap is subtracted once because it was included in both original counts. This same inclusion-exclusion idea can appear in probability form when the question asks for A or B and both can occur.