Compound Probabilities

Deals with the probability of events A and B occurring, where the events are independent.
The probability of A and B is denoted as $P(A \text{ and } B)$ . The "and" symbol resembles a lowercase 'n'.
Formula: $P(A \text{ and } B) = P(A) \times P(B)$
Independent events: The outcome of one event does not affect the outcome of the other.
- Example: Flipping a coin and rolling a die.
- George Washington on the quarter cannot influence the outcome of the die roll.
Example: Probability of flipping a head and rolling a six.
- Probability of flipping a head: $1/2$ (1 desired outcome out of 2 total outcomes).
- Probability of rolling a six: $1/6$ (1 desired outcome out of 6 total outcomes).
- $P(\text{head and six}) = (1/2) \times (1/6) = 1/12$

Mutually exclusive events: Events that cannot occur simultaneously; there is no overlap.
Venn diagrams for mutually exclusive events are disjointed.
The probability of A or B is denoted as $P(A \text{ or } B)$ . The "or" symbol is a capital U.
Formula: $P(A \text{ or } B) = P(A) + P(B)$
Example: Musicians at Ulysses High School.
- 10 saxophonists, 7 flautists, 8 who play no instrument.
- Probability of selecting a student who plays saxophone or flute.
  - Total students: $10 + 7 + 8 = 25$
  - $P(\text{saxophonist or flautist}) = (10 + 7) / 25 = 17/25$

Combined probability deals with events that are not mutually exclusive; they have some overlap.
Venn diagrams for combined events are jointed (they overlap).
Formula: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
- $P(A \text{ and } B)$ represents the overlap between A and B (where both events occur).
Example: Musicians at Lewiston High School.
- 23 trombonists (only), 3 who play both trombone and trumpet, 10 trumpeters (only), 14 who play no instrument.
- Probability of selecting a student who plays trombone or trumpet.
  - Probability of playing trombone: $(23 + 3) / 50 = 26/50$
  - Probability of playing trumpet: $(10 + 3) / 50 = 13/50$
  - Overlap (both trombone and trumpet): $3/50$
  - $P(\text{trombone or trumpet}) = (26/50) + (13/50) - (3/50) = 36/50 = 18/25 = 72\%$
Determining which formula to use:
- Check Venn diagrams; are events separated (mutually exclusive) or overlapping (combined)?
- Independent probability applies when one event's outcome doesn't affect the other.