Interpreting Probability

• Chance behavior: unpredictable in the short run but has a regular and predictable pattern in the long run

• Phenomenon: random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions

• Law of Large Numbers: if we observe more and more repetitions of any change process, the proportion of times that a specific outcome occurs approaches a single value

• Law of Averages/Gamblers Fallacy: occurs when an individual erroneously believes that a certain random event is less likely or more likely to happen based on the outcome of a previous event or series of events

• Simulation: the imitation of chance behavior, based on a model that accurately reflects the situation

Probability Myths:

• The myth of short-run regularity: the idea of probability is that randomness is predictable in the long run

• The myth of the “law of averages”: probability tells us random behavior evens out in the long run

Determining Probability:

• Event: any collection of outcomes from some chance process

• Sample Space S: of a chance process is the set of all possible outcomes

• If two events A and B are mutually exclusive (disjoint) if they have no outcomes in common and so can never occur together (Formula: P(A and B )=)

• If A and B are any two events resulting from some chance process

  (Formula: P(A or B) = P(A) + P(B) − P(A and B)

⭐️ Union means “or” & Intersection means “and”

Formulas to Find Conditional Probability and The General Multiplication Rule:

• Conditional Probability: P(B|A)=P(A and B)/P(A)

• Conditional Probability: P(A|B)=P(A and B)/P(B)

• General Multiplication Rule: P(A and B)= P(A) x P(B|A)

• P(A and B)=P(A|B) x P(B)

• P(A or B)= P(B|A) x P(A)

Independent Vs. Dependent Probability:

• Two events A and B are independent if the occurrence of one event does not change the probability that the other event will happen

• Formulas: P(B|A)=P(A), P(A|B)=P(A)

• If A and B are independent events, then the probability that A and B both occur is P (A ∩ B ) = P (A) * P (B)

Multiplication Rule Example:

During World War II, the British found that the probability that a bomber was lost through enemy action on a mission over occupied Europe was 0.05. Assuming that missions are independent, find the probability that a bomber returned safely from 20 missions.

1-0.05=0.95

0.95^20= 0.3585

The probability that a bomber returned safely from 20 missions is about 35.9%.