33: The Addition Rule

1
3-3 The Addition Rule

• Compound Events
• An event combining 2 or more simple events
• The or Probability problem
• P(A or B) P(A ? B) The probability that A
  occurs, or B occurs, or they both occur we must
  find the number of ways that A can occur and the
  number of ways that B can occur, but in such a
  way that an event is not counted twice.

2
Disjoint / Mutually Exclusive

• Events A B are said to be mutually exclusive if
  they cannot occur simultaneously.
• Example Roll 1 die A roll 2 B roll 5.
• Impossible for a die to show 2 5 at the same
  time mutually exclusive events
• Ex. Roll 1 die A roll 2 B roll even
• Not disjoint since 2 is an even no.

3
Which of the following pairs of events are
mutually exclusive?

• Selecting someone who colors their hair.
• Selecting someone who reads the newspaper.

Since it is possible for a person who colors
their hair to also read the newspaper these
events are Not Disjoint or Mutually Exclusive
4
Which of the following pairs of events are
mutually exclusive?

• Selecting an unmarried television viewer.
• Selecting a TV viewer with an employed spouse.

Disjoint or Mutually Exclusive
5
Which of the following pairs of events are
mutually exclusive?

• Selecting a new car with no defects.
• Selecting a new car with inoperative headlights.

Disjoint or Mutually Exclusive
6
Which of the following pairs of events are
mutually exclusive?

• Reading a book by Mark Twain
• Reading Tom Sawyer

Not Disjoint or Mutually Exclusive
7
Which of the following pairs of events are
mutually exclusive?

• Selecting an ace out of a standard deck.
• Selecting a spade out of a standard deck.

Not Mutually Exclusive
spades
aces
A
8
The Addition Rule

• ? or
• ? and
• P(A ? B) P(A) P(B) if A and B are mutually
  exclusive.
• P(A ? B) P(A) P(B) - P(A ? B) if A and B are
  not mutually exclusive.

9

• Randomly select an individual who is either A
  male or B red
• P(A) 5/15
• P(B) 7/15
• NOT mutually exclusive because some (3) of the
  males are red
• P(A ? B) P(A) P(B) - P(A ? B) both male and
  red
• P(A ? B) 5/15 7/15 - 3/15 (5 7 - 3)/15
  9/15 3/5

10

• Find the probability of randomly selecting a (A
  ) female or (B )seated
• P(A) 10/15
• P(B) 9/15
• P(A ? B both female and seated) 6/15
• P(A or B) 10/15 9/15 - 6/15
• 13/15

11

• Find the probability of randomly selecting a (A
  ) female or (B )green
• P(A) 10/15
• P(B) 8/15
• P(A ? B) 6/15
• P(A or B) 10/15 8/15 - 6/15
• 12/15 4/5

12
Roll 2 Dice

• Find the probability of rolling a 7 or 11.
• A event roll 7
• A 6
• B event roll 11
• B 2
• S 36
• A and B are / are not mutually exclusive?
• P(A?B) 6/36 2/36 8/36 2/9

13
Roll 2 dice

• What is the probability of A rolling 8 or B
  rolling doubles?
• Disjoint events?
• P(A) 5/36
• P(B) 6/36
• P(A?B) 1/36
• P(A?B) 5/36 6/36 - 1/36
• 5/18

14
Titanic Chart p. 130
15
Titanic chart

• Find the probality or randomly selecting one
  person from the Titanic roster, and that person
  being A man, or B boy.
• P(A) 1692/2223
• P(B) 64/2223
• Mutually exclusive events?
• P(A?B) (1692 64)/2223
• 1756/2223

16
Contingency Table
Men Women Boys Girls
Totals Survived 332 318 29 27
706 Died 1360 104 35 18
1517 Total 1692 422
64 56 2223

• Find the probability of randomly selecting a man
  or a boy.

17
Contingency Table
Men Women Boys Girls
Totals Survived 332 318 29 27
706 Died 1360 104 35 18
1517 Total 1692 422
64 56 2223

• Find the probability of randomly selecting a man
  or a boy.
• P(man or boy)

18
Contingency Table
Men Women Boys Girls
Totals Survived 332 318 29 27
706 Died 1360 104 35 18
1517 Total 1692 422
64 56 2223
Mutually Exclusive

• Find the probability of randomly selecting a man
  or a boy.
• P(man or boy) 1692 64 1756 0.790
• 2223 2223 2223

19

• Find P(A man or B survivor)
• P(A) 1692/2223
• P(B) 706/2223
• Mutually exclusive?
• P(man and survivor) 332/2223
• P(A?B) (1692 706 - 332)/2223
• 2066/2223
• Decimal?
• Percent?
• Click Here for more practice with this type
  Smoker Applet

20
15 marbles in an urn

• 3 are red 5 are blue 7 are green.
• What is the probability that I could draw either
  a A red marble or a B green marble?
• P(A) 3/15
• P(B) 7/15
• Mutually exclusive?
• P(A?B) (37)/15 10/15 2/3

21
Complimentary Events

• must be Disjoint!
• Impossible for a ball to be red and not red at
  the same time.

22
Basic Premises
23
Basic premises
24
Find the compliments of the following events

• A coin landing on tails
• Rolling a 4 on one die
• Drawing a heart or spade out of a deck
• Eating at least three cookies
• Rolling a sum less than eleven on two dice

25
Find the indicated probability
Blondes
Brunettes
4 men 6 women
8 men 7 women
P(blonde or red) ?
2 men 3 women
P(man or brunette) ?
Red-heads
P(not red-headed) ?
P(woman or not a blonde) ?