# Section 13.3: Probability Models

## 13.3 Outline

1. Complementary probabilities
1. symbols
2. property of complements
2. Odds
1. in favor
2. against
3. find odds, given the probability
4. find probability, given the odds
3. Conditional probability
1. definition
2. formula
3. procedure for using tree diagrams

## 13.3 Essential Ideas

PROPERTY OF COMPLEMENTS:

P(E) = 1 – P(E compliment)

Odds in favor of an event E :

s/f   (ratio of good to bad)

Odds against an event E :

f/s    (ratio of bad to good)

s = NUMBER OF SUCCESSES
f = NUMBER OF POSSIBILITIES
s + f = n

Suppose that you know P(E ) and wish to find the odds:

odds in favor of an event E :

P(E )/P(E compliment)

odd against on event E :

P(E compliment)/P(E )

Suppose that you know the odds in favor of an event E and wish to find the probability:

P(E ) = s/(s + f)

and

P(E compliment) = f/(s + f)

The fundamental counting principle gives the number of ways of two or more tasks. If task A can be performed in m ways, and if, after task A is performed, a second task B, can be performed in n ways, then task A followed by task B can be performed in mn ways.

The probability of an event E given that another event F has occurred is called a conditional probability, and is denoted by P(E | F).

The procedure for using tree diagrams:

Multiply when moving horizontally across a limb.
Add when moving vertically from limb to limb.
Conditional probabilities; start at their condition.
Unconditional probabilities; start at the beginning of the tree.