## Principles of Probability

• Using Outcomes
• Outcome Distibution
• Complements
• Probability as Counting
• The Rule of Sum
• P(A or B) = P(A) + P(B) - P(A AND B)
• The Product Rule
• P(A and B) = P(A) * P(B)
• For any group of unrelated events, the probability they happen together is the product of their probabilities.
• If the chance of some scenario happening in p, then the probability it happens n times independently is ${p}^{n}$
• Example if you toss a coin, the probability to guess the winning side on the first time is 1/2; two times on the row is 1/4; three times 1/8; 4 times 1/16
• Applying Probability Rules
• If the events have certain dependency and can't happen in the same time, you sum the probabilities
• e.g., flight to be canceled and to be overbooked; these two events are distinct, you can't have a flight that is canceled and overbooked
• If the events are independent and can happen in the same time, you multiply the probabilities
• e.g., one flight can be delayed by wind, second flight can be delayed by traffic; any of these events are independent, so the chance to arrive in time by taking both flights, you need to use the product rule

## Events

• Independent events: The occurrence of one event does not affect the probability of the other event. For example, when flipping a coin, getting “heads” does not change the likelihood of getting “heads” on the next coin flip.
• Dependent events: The occurrence of one event does affect the probability of the other event. For example, if you draw a King from a deck of cards and do not replace it, it causes the probability of drawing another King to decrease.