Probability

You flip a coin and roll a die at the same time. What is the probability of getting heads AND a 6? Your intuition says “pretty low” - and it is: $\frac{1}{2}$ x $\frac{1}{6}$ = $\frac{1}{12}$. Now, what is the probability of getting heads OR a 6 (or both)? That is a different question with a different rule. Confusing “and” with “or” is the most common probability mistake on the MCAT, and it costs easy points.

Basic Probability

The probability of an event is a number between 0 (impossible) and 1 (certain).

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

The probability that an event does NOT happen: P(not A) = 1 - P(A). This complement rule is often the fastest way to solve probability problems.

The AND Rule (Multiplication)

When you want the probability that event A AND event B both occur:

Example: What is the probability that two carrier parents (Aa x Aa) have a child with cystic fibrosis (aa)?

From the Punnett square: P(aa) = $\frac{1}{4}$ for each child.

What is the probability that their first TWO children both have CF? P(aa AND aa) = $\frac{1}{4}$ x $\frac{1}{4}$ = $\frac{1}{16}$. Each birth is independent.

The OR Rule (Addition)

When you want the probability that event A OR event B occurs:

Example: What is the probability of rolling a 2 OR a 5 on a single die? These are mutually exclusive (cannot roll both at once): P = $\frac{1}{6}$ + $\frac{1}{6}$ = $\frac{2}{6}$ = $\frac{1}{3}$.

Example: What is the probability of drawing a heart OR a king from a standard deck? These overlap (king of hearts is both): P = $\frac{13}{52}$ + $\frac{4}{52}$ - $\frac{1}{52}$ = $\frac{16}{52}$ = $\frac{4}{13}$.

Independent vs. Dependent Events

Independent events: The outcome of one does not affect the other. Coin flips, separate births, separate dice rolls. Use P(A and B) = P(A) x P(B).

Dependent events: The outcome of one changes the probability of the other. Drawing cards without replacement, selecting people from a small group. For dependent events: P(A and B) = P(A) x P(B given A).

On the MCAT, genetics problems almost always involve independent events (each child is an independent event). Chemistry and physics problems involving sequential draws or measurements may involve dependent events.

Punnett Squares as Probability Tools

A Punnett square is just a visual multiplication table for gamete probabilities.

For a monohybrid cross Aa x Aa:

Phenotype ratio: $\frac{3}{4}$ dominant, $\frac{1}{4}$ recessive.

For a dihybrid cross (AaBb x AaBb), the probability of any particular combination uses the AND rule across two independent gene loci. P(aabb) = P(aa) x P(bb) = $\frac{1}{4}$ x $\frac{1}{4}$ = $\frac{1}{16}$.

Combinations vs. Permutations

Permutations: Order matters. “How many ways can 3 runners finish a race?” This is nPr = n!/(n-r)!

Combinations: Order does not matter. “How many ways can you choose 3 people from a group of 10?” This is nCr = n!/[r!(n-r)!]

The MCAT rarely requires you to compute factorials. Instead, it tests whether you know which concept to apply. If a genetics problem asks “in how many birth orders can a family of 4 children have exactly 2 boys,” that is a combination: 4C2 = 6 possible orders.

The Complement Shortcut

Sometimes it is easier to calculate the probability of what you do NOT want and subtract from 1.

Example: What is the probability of rolling at least one 6 in four rolls of a die?

Direct calculation is messy (one 6, two 6s, three 6s, four 6s). Instead:

• P(no 6 on a single roll) = $\frac{5}{6}$
• P(no 6 in four rolls) = ($\frac{5}{6}$)⁴ = $\frac{625}{1296}$ = about 0.48
• P(at least one 6) = 1 - 0.48 = 0.52