6. Probability and Randomness Free Online FlashCards

What does a probability model specify?

A probability model specifies the possible outcomes, the events of interest, and the probability assigned to each outcome or event.

How do sample space, outcome, and event differ?

The sample space S is the set of all possible outcomes of a random experiment. An outcome is one particular result; an event is a subset of the sample space.

What two basic constraints must probabilities satisfy?

For every event A, 0 ≤ P(A) ≤ 1, and the entire sample space has probability P(S)=1.

How is an event probability found with equally likely outcomes?

For equally likely outcomes, P(A) equals the number of outcomes in A divided by the number of outcomes in S.

What is the complement rule?

The complement rule is P(A^c)=1−P(A). It is often useful for finding the probability of “at least one” by calculating 1−P(none).

What do intersection and union represent?

The intersection A∩B is the event that both A and B occur. The union A∪B is the event that A, B, or both occur.

What is the general addition rule?

P(A∪B)=P(A)+P(B)−P(A∩B). The intersection is subtracted because adding the two probabilities counts the overlap twice.

What characterizes mutually exclusive events?

Mutually exclusive events cannot occur together, so P(A∩B)=0. Their union therefore has probability P(A)+P(B).

How is conditional probability P(A|B) calculated?

If P(B)>0, conditional probability is P(A|B)=P(A∩B)/P(B). The condition B restricts attention to outcomes in B.

Why is the denominator 96 in the fitness-app example?

In the fitness-app table, P(regular exercise|app use)=72/96=0.75. The denominator is 96, because the condition restricts attention to app users.

What is the multiplication rule?

The multiplication rule is P(A∩B)=P(B)P(A|B), or equivalently P(A)P(B|A). It calculates the probability that both events occur.

When are two events independent?

Events are independent when knowing that one occurred does not change the probability of the other. Equivalently, P(A∩B)=P(A)P(B).