Download Disjoint events cannot be independent

Warm-Up:
1) Define sample space.
2) Give the sample space for the sum of the numbers for a pair of
dice.
3)
You flip four coins. What’s the probability of getting exactly two
heads? (Hint: List the outcomes first).
4)
Joey is interested in investigating so-called hot streaks in foul
shooting among basketball players. He’s a fan of Carla, who has
been making approximately 80% of her free throws. Specifically,
Joey wants to use simulation methods to determine Carla’s longest
run of baskets on average, for 20 consecutive free throws.
a) Describe a correspondence between random digits from a
random digit table and outcomes.
b) What will constitute one repetition in this simulation?
c) Starting with line 101 in the random digit table, carry out 4
repetitions and record the longest run for each repetition.
d) What is the mean run length for the 4 repetitions?
P(A U B) = P(A) + P(B) => “A or B”
“A union B” is the set of all outcomes that are either in A or in B.
Disjoint/Complement
Probability Distribution
Age group
(yr)
Probability
18-23
24-29
30-39
40+
.57
.17
.14
.12
1) Find the probability that the student is not in the
traditional undergraduate age group of 18-23
2) Find P(30+ years)
Venn Diagram
Find P(A), P(B), P(C)
Find P(A’), P(B’), P(C’)
Venn Diagram: Union
(“Or”/Addition Rule)
Find:
1) P(AUB) =getting an even
number or a number greater
than or equal to 5 or both
2) P(AUC) =getting an even
number or a number less
than or equal to 3 or both
3) P(BUC)=getting a number
that is at most 3 or at least 5
or both.
Remember this?
Ex. 6.14, p. 420
• Because all 36 outcomes together must have
probability 1 (Rule 2), each outcome must have
probability 1/36.
• Now find P(rolling a 7) =
1st
digit
1
Prob. .301
2
3
4
5
6
7
8
9
.176
.125
.097
.079
.067
.058
.051
.046
Consider the events A = {first digit is 1}, B = {first digit is 6 or
greater}, and C = {a first digit is odd}
a) Find P(A) and P(B)
b) Find P(complement of A)
c) Find P(A or B)
d) Find P(C)
e) Find P(B or C)
Ex. 6.16,
p. 422
1st
digit
1
Prob. 1/9
2
3
4
5
6
7
8
9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
Find the probability of the event B that a randomly
chosen first digit is 6 or greater.
• The probability that
BOTH events A and B
occur
• A and B are the
overlapping area common
to both A and B
• Only for INDEPENDENT
events
Example
If the chances of success for surgery A are
85% and the chances of success for surgery
B are 90%, what are the chances that both
will fail?
Venn Diagram: Intersection
(“And”/* Rule)
Find:
1) P(A and B) =getting an even
number that is at least 5
2) P(A and C) =getting an even
number that is at most 3
3) P(B and C)=getting a number
that is at most 3 and at least 5.
Finding the probability of “at least one”
P(at least one) = 1-P(none)
Many people who come to clinics to be tested for HIV don’t
come back to learn the test results. Clinics now use “rapid HIV
tests” that give a result in a few minutes. Applied to people who
don’t have HIV, one rapid test has probability about .004 of
producing a false-positive. If a clinic tests 200 people who are
free of HIV antibodies, what is the probability that at least one
false positive will occur?
N = 200
P(positive result) =.004, so P(negative result)=1-.004=.996
Big Picture
• + Rule holds if A and B are disjoint/mutually
exclusive
• * Rule holds if A and B are independent
• * Disjoint events cannot be independent!
Mutual exclusivity implies that if event A
happens, event B CANNOT happen.
Conditional probability: Pre-set
condition (“given”)
Find:
1) P(A given C) =getting an
even number GIVEN that
the number is at most 3.
2) P(A given B) =getting an
even number GIVEN that
the number is at least 5.
In building new homes, a contractor finds that the
probability of a home-buyer selecting a two-car garage
is 0.70 and selecting a one-car garage is 0.20. (Note
that the builder will not build a three-car or a larger
garage).
1) What is the probability that the buyer will select either
a one-car or a two-car garage?
2) Find the probability that the buyer will select no
garage.
3) Find the probability that the buyer will not want a twocar garage.