Events Involving Not and Or; Odds

The probability of an event E not occurring is 1-P(E)

Example: Toss a die

    What is the probability of a 3 ?
    1/6
    What is the probability of not rolling a 3?
    1-1/6 = 5/6

Sometimes it is easier to compute probability this way.

If you select a card from a standard deck of cards, what is the probability it is not a red picture card?

     There are 6 red picture cards
     P(Red picture Card) = 6/52 = 3/26
     P(Not a red picture Card) = 1-3/26 = 23/26

Two events are mutually exclusive if it is impossible for them to occur simultaneously.
Selecting a King and a 4 from a deck of cards by drawing 1 card.
Rolling two dice, a 1 shows and the sum is 11
If two events, A and B, are mutually exclusive, the probability that one or the other occurs is the sum of their individual probabilities. P(A or B) = P(A) + P(B) if A and B are mutually exclusive.

Example: What is the probability of rolling a 7 or 11 when two dice are rolled?

    Choices for 7 are {1-6,2-5,3-4,4-3,5-2,6-1}
    n(7) = 6
    Choices for 11 are {5-6, 6-5}
    n(11)  = 2
    n(S) = 6×6 = 36

Problem 22, page 6-2

    n(S) = 67
    P(R) = 30/67
    P(D) = 25/67
    P(I) = 8/67
    P(G) = 4/67
    P(D or G) = 25/67 + 4/67 = 29/67

If the events are not mutually exclusive then
P(A or B) = P(A) + P(B) - P(A and B)

Example Roll two 6 sided dice

Find the probability of rolling a 2 on the first die or the sum is a 7
P(2) = 6/36
P(7) = 6/36
P(2 on the first die and the sum is 7) = 1/36
P(2 or 7) = 11/36

Example Number 26

P(5) = 4/52 (or 1/12)
P(black) = 13/52 = 1/4
P(black 5) = 2/52
P(5 or black) = 4/52  + 13/52 - 2/52 
              = 15/52

Do numbers 40,42, 44

P(not completed four years of high school) = 29/174
P(less than 4 years or four years) = 29/174+56/174 = 85/174
P(4 years or is a woman) = 29/174+92/174-15/174 = 106/174

Odds
Odds can be in favor of an event happening
Odds can also be against an event happening
Unless otherwise stated, given odds are against an event happening.
The odds in favor of an event E = P(E)/P(not E)
The odds against an event E = P(not E)/P( E)
Odds are expressed as a fraction or in the form a:b, or a to b

Example: A six sided die is rolled.

Find the odds in favor of rolling a 5 or a 6
P(5) = 1/6
P(6) = 1/6 
P(5 and 6) = 0

P(5 or 6) = 2/6 or 1/3
P(not 5 or 6) = 1-1/3 = 2/3

Odds in favor of a 5 or a 6 are  1/3 /  2/3  or 1/2 or 1:2 or 1 to 2
Odds against a 5 or a 6 are 2/3 / 1/3 or 2/1 or 2:1 or 2 to 1

Notice we can recover the probability of an event from the odds:

The odds against a 5 or a 6 are 2:1 
So P(E) = 1/(2+1)
   P(not E) = 2/(2+1)

Example number 66
```
P(E) = 1/4
P(not E) = 3/4

odds 1:3
```

Homework

Please do 1-60 odd.