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MAT-144 · Topic 06 · Cheat Sheet

Probability

The formulas, the moves, and the traps for Topic 6.

Key formulas

Every Topic 6 problem reduces to one of these moves. The hardest part is reading the prompt carefully enough to know which one.

Simple probability
P(event) = favorable outcomestotal outcomes
Always a value between 0 and 1 (or 0% and 100%). Both top and bottom must count the same kind of thing — cards, rolls, marbles, whatever the sample space is.
Fundamental counting principle
total outcomes = n1 × n2 × n3 × …
When a choice is made in stages, multiply the options at each stage. 4 entrées × 3 sides × 2 drinks = 24 meals. The most flexible counting tool in the topic.
Addition rule (A or B)
General:  P(A or B) = P(A) + P(B) − P(A and B)
Mutually exclusive:  P(A or B) = P(A) + P(B)
Subtract the overlap so you don't double-count. “Mutually exclusive” just means no overlap (A and B can't both happen), so the subtraction term is zero.
Permutations (order matters)
nPr = n!(n − r)!
Use when arranging r items out of n in a specific order — finish places in a race, the first three speakers at a panel, three-letter passwords. Excel: =PERMUT(n, r).
Combinations (order doesn't matter)
nCr = n!r! (n − r)!
Use when choosing r items from n where the order doesn't matter — a hand of cards, a committee of 4, a pizza topping combination. Excel: =COMBIN(n, r).
Expected value
E(X) = Σ (value × probability)
For every possible outcome, multiply the dollar value by the probability, then sum. A “fair” game has E(X) = $0. Carnival games, insurance policies, and lottery tickets all have negative expected value for the customer.
Complement
P(not A) = 1 − P(A)
When “at least one” gets messy, compute the probability of the opposite (“none”) and subtract from 1. Especially powerful with independent trials: P(at least one head in 4 flips) = 1 − (0.5)4.

Common mistakes

Most Topic 6 misses come from misreading the prompt, not from arithmetic. These are the lines to slow down on.

  1. Permutation vs combination mixed up. “In how many orders” / “arrange” / “first, second, third” → permutation. “How many groups” / “choose” / “committee” → combination. Read the prompt once for vocabulary before computing.
  2. Forgot the overlap in “A or B.” If a card is “red or a king,” you have to subtract the two red kings, otherwise they're counted twice.
  3. Added when you should have multiplied. “Pick a shirt and a pair of pants” uses the counting principle (multiply). “Pick a shirt or a pair of pants” uses the addition rule.
  4. Treated dependent events as independent. Drawing two cards without replacement changes the second probability (51 cards left, not 52). Drawing with replacement keeps both at 1/52.
  5. Listed outcomes carelessly in a sample space. Two coin flips: HH, HT, TH, TT — four outcomes, not three. HT and TH are different outcomes even though the count is the same.
  6. Expected value with sign errors. If the game costs $5 to play, the “value” column has net dollars: a $20 prize is +$15, a loss is −$5. Don't forget to subtract what you paid in.
  7. Forgot 0! = 1. Both 0! and 1! equal 1. nPn = n!, and nCn = 1. Excel handles this correctly; by hand, watch out.
  8. Reported a probability greater than 1. Any answer that comes out > 1 or < 0 is wrong. Probabilities are bounded between 0 and 1; if your “answer” is 1.4, you double-counted or set up the fraction backwards.
  9. Theoretical vs experimental conflated. Theoretical = computed from the structure (P(heads) = 1/2). Experimental = measured from data (53 heads out of 100 flips = 0.53). Both are valid; they answer different questions.
  10. Forgot to reduce the fraction. 6/12 is a correct probability but ALEKS usually wants 1/2. Always reduce at the end.

Quick reference

Sample spaces worth memorizing, the permutation-vs-combination decision tree, and the Excel calls.

Sample spaces to know cold
One coin flip  →  2 outcomes (H, T)
One die roll  →  6 outcomes (1…6)
Two coin flips  →  4 outcomes (2 × 2)
Two dice  →  36 outcomes (6 × 6)
A standard deck  →  52 cards (4 suits × 13 ranks)
Permutation or combination?
Does order matter? Try rearranging your top picks — if a different order means a different answer, it's a permutation. If the same group counts as one regardless of order, it's a combination.

Lottery numbers? Combination. Race finishers? Permutation. Pizza toppings? Combination. Locker codes? Permutation (and repetition is usually allowed, so the counting principle is faster).
Probability vocabulary
Outcome — a single possible result.
Sample space — the set of all outcomes.
Event — a subset of the sample space we care about.
Mutually exclusive — two events that can't both happen.
Independent — the outcome of one doesn't affect the other.
Complement — everything in the sample space except the event.
Excel for probability
=PERMUT(n, r)  permutations
=COMBIN(n, r)  combinations
=FACT(n)  n factorial
=SUMPRODUCT(values, probs)  expected value in one shot
=RAND()  random number 0–1, for simulating experiments
Theoretical vs experimental
Theoretical — computed from the structure of the problem. P(heads) = 1/2 = 0.5.
Experimental — measured from observed trials. 53 heads in 100 flips = 0.53.
The Law of Large Numbers: experimental converges to theoretical as the number of trials grows.
MAT-144 · Topic 06 · Cheat Sheet · mat144.com