Complements and Conditional Probability

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1.

The complement of an event A is the event that A does not occur, denoted as A'. The probability of the complement is given by P(A') = 1 - P(A).

The probability of an event A is calculated as P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}.

Conditional probability is the probability of an event A given that event B has occurred, denoted as P(A|B) = \frac{P(A \cap B)}{P(B)}.

The probability of at least one event occurring is given by P(at least one) = 1 - P(none).

The probability that a randomly selected bulb is not defective is 95%, or P(not defective) = 0.95.

For two independent events A and B, the probability of both A and B occurring is P(A \cap B) = P(A) \times P(B).

P(not in club) = \frac{138}{378}.

P(Latin | in club) = \frac{41}{240}.

The total probability rule states that if events B1, B2, ..., Bn form a partition of the sample space, then P(A) = \sum_{i=1}^{n} P(A | B_i) P(B_i).