### P(A and B)

```The following table shows the number of people that like a
particular fast food restaurant.
McDonald’s
Burger
King
Wendy’s
Male
20
15
10
Female
20
10
25
1) What is the probability that a person likes Wendy’s?
7/20
2) What is the probability that a person is male given they like
Burger King?
3/5
3. What is the probability that a randomly chosen person is
female or likes McDonald’s?
3/4
CCGPS Geometry
UNIT QUESTION: What connection
does conditional probability have to
independence?
Standard: MCC9-12.S.CP.1-7
Today’s Question:
How can I determine if 2 events are
independent of each other?
Standard: MCC9-12.S.CP.1, 7
Probability
Independent vs.
Dependent events
Independent Events
• Two events A and B, are independent if
the fact that 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)
Experiment 1
• A coin is tossed and a 6-sided die is
rolled. Find the probability of landing
on the head side of the coin and
rolling a 3 on the die.
P(3) = 1/6
= 1/2  1/6
= 1/12
Experiment 2
• A card is chosen at random from a
deck of 52 cards. It is then replaced
and a second card is chosen. What
is the probability of choosing a jack
and an eight?
P (jack) = 4/52
P (8) = 4/52
P (jack and 8) = 4/52  4/52
= 1/169

Experiment 3
• 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

Experiment 4
• A school survey found that 9 out of 10
students like pizza. If three students are
chosen at random with replacement,
what is the probability that all three
students like pizza?
P (student 1 likes pizza) = 9/10
P (student 2 likes pizza) = 9/10
P (student 3 likes pizza) = 9/10
P (student 1 and student 2 and student 3 like
pizza) = 9/10  9/10  9/10 = 729/1000

Dependent Events
• Two events A and B, are dependent if the
fact that 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)
Experiment 1
• 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 without replacing the first one. What is
the probability of choosing a green and a
yellow marble?
P (green) = 5/16
P (yellow given green) = 6/15
P (green and then yellow) = P (green)  P (yellow)
= 1/8
Experiment 2
• An aquarium contains 6 male goldfish and 4
female goldfish. You randomly select a fish
from the tank, do not replace it, and then
randomly select a second fish. What is the
probability that both fish are male?
P (male) = 6/10
P (male given 1st male) = 5/9
P (male and then, male) = 1/3

Experiment 3
• 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?

Independent vs.
Dependent
Determining if 2 events
are independent
Independent Events
• Two events are independent if the
following are true:
P(A|B) = P(A)
P(B|A) = P(B)
P(A AND B) = P(A) ⋅ P(B)
• To show 2 events are independent, you
must prove one of the above
conditions.
Experiment 1
• Let event G = taking a math class. Let
event H = taking a science class. Then,
G AND H = taking a math class and a
science class.
• Suppose P(G) = 0.6, P(H) = 0.5, and
P(G AND H) = 0.3.
• Are G and H independent?
P(G  H )  P(G)  P( H )?
0.3  0.6  0.5
Experiment 2
• In a particular college class, 60% of the students
are female. 50% of all students in the class have
long hair. 45% of the students are female and
have long hair. Of the female students, 75% have
long hair.
• Let F be the event that the student is female. Let L
be the event that the student has long hair.
• One student is picked randomly. Are the events
of being female and having long hair
independent?
P( F  L)  P( F )  P( L)?
0.45  0.6  0.5
0.45  0.3
```