Probability - ACT Math

$\frac{1}{2}$. One of two equally likely outcomes on a fair coin.

$0$. An event that cannot occur has probability 0.

$P(A\cap B)=P(B\cap A)$. Intersection is commutative - order doesn't matter.

$P((A\cup B)^c)=1-P(A\cup B)$. Complement rule applied to the union of events.

$n_1\cdot n_2\cdots n_k$. Multiply choices at each step for total possibilities.

$\frac{1}{2}$. Two outcomes HT and TH out of four total possibilities.

$\frac{1}{6}$. Six ways to get sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

Events that cannot occur at the same time. No overlap between the events is possible.

1. Impossible events can never happen.

1. Certain events always happen.

1. All possible outcomes must account for the entire sample space.

$\frac{1}{6}$. One favorable outcome (rolling 3) out of 6 possible outcomes.

$\frac{1}{2}$. One favorable outcome (heads) out of 2 possible outcomes.

$\frac{1}{4}$. 13 hearts out of 52 total cards.

$\frac{1}{13}$. 4 aces out of 52 total cards.

$\frac{1}{2}$. Three even numbers (2, 4, 6) out of 6 possible outcomes.

Two events are complementary if their probabilities sum to 1. One event is the complement of the other.

$P(A') = 1 - P(A)$. The complement probability equals 1 minus the original probability.

$\frac{5}{6}$. Complement of rolling a 5, which has probability $\frac{1}{6}$.

$P(A \text{ or } B) = P(A) + P(B)$. Add probabilities when events cannot occur simultaneously.

$\frac{1}{3}$. Two favorable outcomes (2 or 5) out of 6 possible outcomes.

Events are independent if the occurrence of one does not affect the other. One event's outcome doesn't influence the other.

$P(A \text{ and } B) = P(A) \times P(B)$. Multiply probabilities when events don't affect each other.

$\frac{1}{4}$. $\frac{1}{2} \times \frac{1}{2}$ for two independent coin flips.

$\frac{1}{221}$. $\frac{4}{52} \times \frac{3}{51}$ for dependent events without replacement.

The probability of an event given that another event has occurred. Probability changes based on prior information.

$\frac{4}{51}$. 4 queens remain out of 51 cards after drawing a king.

$\frac{7}{13}$. 26 red cards plus 13 clubs minus 13 overlapping red clubs.

$\frac{1}{6}$. 6 ways to roll sum of 7 out of 36 total outcomes.