Download probabilities of compound events

Probability of Compound Events
Vocabulary
compound event combines two or more events, using
the word and or the word or.
The word “or” in probability means Union  of two
events
The word “and” in probability means the intersection 
of two events
mutually exclusive have no common outcomes.
P(A B) = 0
Overlapping events have at least one common
outcome.
Mutually Exclusive Events
The probability is found by summing the individual
probabilities of the events:
P(A  B) = P(A) + P(B)
A Venn diagram is used to show mutually exclusive events.
Mutually Exclusive Events
Find the probability that a girl’s favorite department store is
0.45
Macy’s or Nordstrom.
Find the probability that a girl’s favorite store is not JC Penney.
0.90
Macy’s
Saks
Nordstrom
JC Penney
Bloomingdale’s
0.25
0.20
0.20
0.10
0.25
Mutually Exclusive Events
When rolling two dice, what is probability that your sum will be
4 or 5?
1&3, 2&2, 3&1
Possibilities sum of 4 _____________________________
1&4, 2&3, 3&2, 4&1
Possibilities sum of 5 _____________________________
Total possible combinations of rolling 2 die ____________
36
P(sum4  sum 5) = P(sum5) + P(sum4)
7/36
Mutually Exclusive Events
What is the probability of picking a queen or an ace from a
deck of cards
P(Ace) = 4/52
P(QN) = 4/52
P(AUQ) = 8/52
= 2/13
Overlapping Events
Probability that overlapping events A or B will
occur expressed as:
P(M  E) = P(M) + P(E) - P(ME)
Overlapping Events
Find the probability of picking a king or a club in a deck of
cards.
Kings____
4
13
Clubs ____
Kings that
1
are clubs ____
52
Total Cards ____
P(KC) = P(K) + P(Clubs) – P(kings that are clubs)
P(KC) = 4/52 + 13/52 – 1/52 =
16/52= 4/13
Overlapping Events
Find the probability of picking a female or a person from
Tennessee out of the 31 committee members.
21
Females ____
12
People from TN ____
8
Females from TM ____
31
Total People _____
Fem
Male
TN
8
4
AL
6
3
GA
7
3
21 12 8 25
  
31 31 31 31
Independent Events
• Two events A and B, are independent if A occurs
& does not affect the probability of B occurring.
• Examples- Landing on heads from two different
coins, rolling a 4 on a die, then rolling a 3 on a
second roll of the die.
• Probability of A and B occurring:
P(A and B) = P(A) ∙ P(B)
• A jar contains three red, five green, two
blue and six yellow marbles. A marble is
chosen at random from the jar. After
replacing it, a second marble is chosen.
What is the probability of choosing a green
and a yellow marble?
P (green) = 5/16
 P (yellow) = 6/16
 P (green and yellow) = P (green) ∙ P (yellow)
= 15 / 128

Dependent Events
• Two events A and B, are dependent if A occurs &
affects the probability of B occurring.
• Examples- Picking a blue marble and then picking
another blue marble if I don’t replace the first one.
• Probability of A and B occurring:
P(A and B)=P(A) ∙ P(B given A)
• A random sample of parts coming off a
machine is done by an inspector. He found
that 5 out of 100 parts are bad on average.
If he were to do a new sample, what is the
probability that he picks a bad part and then
picks another bad part if he doesn’t replace
the first?
 P (bad) = 5/100
 P (bad given bad) = 4/99
 P (bad and then bad) = 1/495
• A jar contains three red, five green, two
blue and six yellow marbles. A marble is
chosen at random from the jar. A second
marble is chosen. What is the probability of
choosing a green and a yellow marble if the
first marble is not replaced?
 P (green) = 5/16
 P (yellow) = 6/15
 P (green and yellow) = P (green) ∙ P (yellow)
= 30 / 240 = 1/8
• A jar contains three red, five green, two
blue and six yellow marbles. A marble is
chosen at random from the jar. A second
marble is chosen. What is the probability of
choosing a green marble both times if the
first marble is not replaced?
 P (green) = 5/16
 P (green) = 4/15
 P (green and green) = P (green) ∙ P (green)
= 20 / 240 = 1/12
P(A or B) = P(A) + P(B)
-Drawing a king or a queen
-Selecting a male or a female
-Selecting a blue or a red marble
P(A and B) = P(A) ∙ P(B)
P(A or B) = P(A) + P(B) - P(overlap)
-Drawing a king or a diamond
-rolling an even sum or a sum
greater than 10 on two dice
-Selecting a female from Georgia
or a female from Atlanta
P(A and B) = P(A) ∙ P(B given A)
WITH REPLACEMNT:
WITHOUT REPLACEMENT:
-Drawing a king and a queen
-Drawing a king and a queen
-Selecting a male and a female
-Selecting a male and a female
-Selecting a blue and a red marble
-Selecting a blue and a red marble
Find Probabilities of Compound Events
Example 1 Find the probability of A or B
You randomly choose a card from a
standard deck of 52 playing cards.
a. Find the probability that you choose a 9 or a
King.
b. Find the probability that you choose an Ace or a
Solution
spade.
a. Choosing a 9 or a King are mutually exclusive events.
P(9 or King)  P9  PKing 
4
4


52
52
________
8
2


52
13
________
________
________
Find Probabilities of Compound Events
Example 1 Find the probability of A or B
You randomly choose a card from a
standard deck of 52 playing cards.
a. Find the probability that you choose a 9 or a
King.
b. Find the probability that you choose an Ace or
a spade. Solution
b. Because there is an Ace of spades, choosing an Ace
or spade are ___________________.
There are 4
overlapping events
Aces, 13 spades, and 1 Ace of spades.
P(Ace or spade)  PAce  Pspade  PAce and spade

4
13
1



52
52
52
________
________
________
16 
52
________
4
________
13
Find Probabilities of Compound Events
Example 2 Find the probability of A and B
You roll two number cubes. What is the
probability that you roll a 1 first and a 2
second?
Solution
independent The number on one number cube does not
The events are _____________.
affect the other.
P(2)
P(1)  ____
P(1 and 2)  ____

1
1

6
6
________
________

1
36
________
Find Probabilities of Compound Events
Example 3 Find the probability of A and B
Markers A box contains 8 red markers
and 3 blue markers. You choose one
marker at random, do not replace it, then
choose a second marker at random. What
is the probability that both markers are
blue?
Solution
dependent Before
Because you do not replace the first marker, the events are __________.
you choose a marker, there are 11 markers, 3 of them are blue. After you choose a
blue marker, there are 10 markers left and two of them are blue. So, the
______________________
conditional
probability that the second marker is blue given that the first
marker is blue, is
3
10
Find Probabilities of Compound Events
Example 3 Find the probability of A and B
Markers A box contains 8 red markers
and 3 blue markers. You choose one
marker at random, do not replace it, then
choose a second marker at random. What
is the probability that both markers are
blue? Solution
P(blue given____
blue)
P(blue and then blue) P(blue)
_____  __________

2
3


10
11 ________
________
3
55
________
Find Probabilities of Compound Events
1. In a standard deck of cards, find
the probability you randomly
select a King of diamonds or a
spade.
Choosing a King of diamonds or a spade
are mutually exclusive events.
P(King of diamonds or a spade) 
P(King of diamonds)  P(spade)
13
1
7
14



52 52 52 26
Find Probabilities of Compound Events
2. In Example 3, suppose there are also
4 orange markers in the box.
Calculate the probability of selecting
a blue marker and then an orange
marker, without replacement.
P(blue and then orange) 
P(blue) + P(orange given blue)
4
3
1 2 17

  
15 14 5 7 35