Download Probability Rules and Compound Events

4.2 Compound Events
*Probability of A AND B*
What does AND mean?
Can you be…
A boy AND a girl
A Freshman AND a girl
A Senior AND a Freshman
A grasshopper AND an insect
SpongeBob & Papa Smurf
Yes
No
In probability AND means:
+ 2 events occurring together
+ The overlap of 2 events
+ When one thing satisfies BOTH events
I. Independent Events
Independent Events: Two events are independent if the outcome
of one event does not change the probability of the second event.
Dependent Events: Event # 1 is a dependent event where the
outcome of the first event changes the outcome of the second event
Independent Events
Have no effect on each other
Don’t change each other’s
probability
Replacement
Dependent Events
The first event effects the 2nd
The probability of the 2nd is
different now that the first
happened
Without Replacement
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Event 1
Rolling a 6 on
one die
Pick winning
horse in first
race
Pick winning
horse in first
race
Draw an Ace
(without
replacing)
Event 2
Rolling a 6 on a
second die
Pick winning horse
in second race
Independent Dependent
X
X
Pick second place
horse in SAME race
X
Draw another Ace
X
II. Probability of Independent Events
P (A and B) = P(A) * P(B)
…only works if A and B are INDEPENDENT events
Example:
You roll a pair of dice:
a) What is the probability of rolling two 4’s
P(4 and 4) = P(4)*P(4) = 1/6 * 1/6 = 1/36
b) What is the probability of rolling an EVEN # on die 1 and an
ODD # on die 2?
P(Even and Odd) = P(Even)*P(Odd) = ½ * ½ = 1/4
III. Conditional Probability
Conditional Probability: you calculate the probability of an
event assuming a certain related event has just happened.
Ex: Drawing 2 cards. Your first is an Ace, what is the probability
that the second will be a King?
P(King, given Ace)  now there are only 51 cards left = 4/51
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Ex: You have a bag of 50 red and 50 green M&Ms. You pick out
a green M&M and eat it.
a) What is the probability that the second will be Red?
P(Red, given Green) = 50/99
b) What is the probability that the second will be Green?
P(Green, given Green) = 49/99
IV. Probability of Dependent Events
P(A and B) = P(A) * P(B, given that A has occurred)
P(A and B) = P(B) * P(A, given that B has occurred)
How can you tell if two events are DEPENDENT?
P(A) ≠ P(A, given B) and P(B) ≠ P(B, given A)
How can you tell if two events are INDEPENDENT?
P(A) = P(A, given B) and P(B) = P(B, given A)
Example:
You have a bag of six marbles, three blue and three red. You
pick a marble from the bag and then, without replacement, you
draw a second marble. What is the probability of getting two
reds?
1.) What is the P(R)? If you get a Red on the first draw, what is
the P(R,given R on first) *red on the second?
P(R) = ½ P(R, given R) = 2/5
because there are only 5 marbles left.
2.) Compute the probability using the multiplication rule for
these dependent events.
P(R,R)= 1 * 2  1
2 5
5
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V. EXAMPLES
1.) At the University of Kentucky, a random selection of 140
people was asked what political party they associated themselves
with. The people were grouped into two categories, students or
professors. For notation purposes, we will use S(student),
P(professor), D(democrat), R(republican), I(independent).
Suppose a person was selected at random out of the 140, find the
following probabilities:
Person
Democrat Republican Independent Row Total
Type
(D)
(R)
(I)
Professor
5
34
9
48
Student
63
21
8
92
Column
68
55
17
140 *Grand
Total
Total
a) Compute P(D) and P(P).
68
48
P(D) =
≈0.486 P(P) =
≈0.343
140
140
b) Compute P(D, given P)
P(D, given P) =
5
# Democratic Pr ofessors
=
≈ 0.104
48
# Pr ofessors
c) Are the events Democrat and Professor independent?
Does P(D) = P(D, given P)?
0.486 ?=?0.104
*NO!!! Therefore, the probability of being a Democrat
depends on whether or not you are a Professor.
d) Compute P(D and P)
P(D and P) = P(P) * P(D, given P)
48 5
5


 0.036
140 48 140
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