Refresher - Independence and Mutual Exclusivity

Definition: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring.

Some other examples of independent events are:

  • Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
  • Choosing a marble from a jar AND landing on heads after tossing a coin.
  • Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card.
  • Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.

To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, and then multiply the probabilities. This multiplication rule is defined symbolically below. Note that multiplication is represented by AND.

 

More on Independence
Consider two events which might occur in succession, such as two flips of a coin. If the outcome of the first event has no effect on the probability of the second event, then the two events are called independent. For two coin flips, the probability of getting a "head" on either flip is 1/2, regardless of the result of the other flip.

Multiplication Rule:
When two events, A and B, are independent, the probability of both occurring is
P(A⋂B)=P(A).P(B)

Rule of Addition
If events A and B come from the same sample space, the probability that event A and/or event B occur is equal to the probability that event A occurs plus the probability that event B occurs minus the probability that both events A and B occur.
P(A⋃B)=P(A)+P(B)-P(A⋂B)

Note: Invoking the fact that P( A ∩ B ) = P( A )P( B | A ), the Addition Rule can also be expressed as
P(A⋃B)=P(A)+P(B)-P(A)P(B|A)

 

DEF: If two events have no outcomes in common, then they are called disjoint or Mutually Exclusive. For example, the possible outcomes of picking a single marble are disjoint: only one color is possible on each pick. The addition of probabilities for disjoint events is the third basic rule of probability:

If two events A and B are disjoint, then the probability of either event is the sum of the probabilities of the two events:

P(AorB)=P(A⋃B)=P(A)+P(B)

In other words, two events are mutually exclusive if they cannot occur at the same time If A and B are not mutually exclusive, then
P(A⋃B)=P(A)+P(B)-P(A⋂B)
P(AorB)=P(A)+P(B)-P(AandB)

The chance of any (one or more) of two or more events occurring is called the union of the events. The probability of the union of disjoint events is the sum of their individual probabilities.

For example, the probability of drawing either a blue, purple, or silver marble from a bowl of seven differently colored marbles is the sum of the probabilities of drawing any of these marbles:17+17+17=37.