Mutually exclusivity and introduction to conditional probability and medical testing.

1. Medical Testing
or
Why are doctors so
darn cagey?!?
Let me check your sugar. by flickr user Cataract eye

2. The probability that Gallant Fox will win the first race is 2/5 and that
Nashau will win the second race is 1/3.
1. What is the probability that both horses will win their
respective races?
2. What is the probability that both horses will lose their
respective races?
3. What is the probability that at least one horse will win a race?

3. In a 54 person sudden death tennis tournament how many games
must be played to determine a winner?

4. Chad has arranged to meet his girlfriend, Stephanie, either in the
library or in the student lounge. The probability that he meets her
in the lounge is 1/3, and the probability that he meets her in the
library is 2/9.
a. What is the probability that he meets her in the library or
lounge?
b. What is the probability that he does not meet her at all?

5. Mutually Exclusive Events ...
Two events are mutually exclusive (or disjoint) if it is impossible for
them to occur together.
Formally, two events A and B are mutually exclusive
if and only if
Mutually Exclusive

6. Mutually Exclusive Events ...
Two events are mutually exclusive (or disjoint) if it is impossible for
them to occur together.
Formally, two events A and B are mutually exclusive
if and only if
Mutually Exclusive
Examples:
1. Experiment: Rolling a die once
Sample space S = {1,2,3,4,5,6}
Events A = 'observe an odd number' = {1,3,5}
B = 'observe an even number' = {2,4,6}
A B = (the empty set), so A and B are mutually exclusive.

7. Mutually Exclusive Events ...
Two events are mutually exclusive (or disjoint) if it is impossible for
them to occur together.
Formally, two events A and B are mutually exclusive
if and only if
Mutually Exclusive Not Mutually Exclusive
Examples:
1. Experiment: Rolling a die once
Sample space S = {1,2,3,4,5,6}
Events A = 'observe an odd number' = {1,3,5}
B = 'observe an even number' = {2,4,6}
A B = (the empty set), so A and B are mutually exclusive.
2. A subject in a study cannot be both male and female, nor can
they be aged 20 and 30. A subject could however be both male
and 20, or both female and 30.

8. Probability of non-Mutually Exclusive Events ...
Example
Suppose we wish to find the probability of drawing either a king or a spade
in a single draw from a pack of 52 playing cards.
We define the events A = 'draw a king' and B = 'draw a spade'
Since there are 4 kings in the pack and 13 spades, but 1 card is
both a king and a spade, we have: Not Mutually Exclusive
P(A U B) = P(A) + P(B) - P(A B)
= 4/52 + 13/52 - 1/52
= 16/52
So, the probability of drawing either a king or a spade is 16/52 = 4/13.

9. Identify the events as:
dependent mutually exclusive
Drag'n Drop
Baby! independent not mutually exclusive
a. A bag contains four red and seven black marbles. The event
is randomly selecting a red marble from the bag, returning it to
the bag, and then randomly selecting another red marble from the
bag. independent mutually exclusive
b. One card - a red card or a king - is randomly drawn from a
deck of cards. n/a not mutually exclusive
c. A class president and a class treasurer are randomly
selected from a group of 16 students. dependent mutually exclusive
d. One card - a red king or a black queen - is randomly drawn
from a deck of cards. n/a mutually exclusive
e. Rolling two dice and getting an even sum or a double.
independent not mutually exclusive

10. Probabilities involving quot;andquot; and quot;orquot; A.K.A quot;The Addition Rulequot;...
The addition rule is a result used to determine the probability that
event A or event B occurs or both occur.
The result is often written as follows, using set notation:
Not Mutually Exclusive
P(A or B) = P(A B) = P(A)+P(B) - P(A B)
where:
P(A) = probability that event A occurs
P(B) = probability that event B occurs
P(A U B) = probability that event A or event B occurs
P(A B) = probability that event A and event B both occur
P(A and B) = P(A B) = P(A)*P(B)

11. Suppose a test for cancer is known to be 98% accurate. This means
that the outcome of the test is correct 98% of the time. Suppose that
0.5% of the population have cancer. What is the probability that a
person who tests positive for cancer has cancer?
Suppose 1 000 000 randomly selected people are tested. There are four
possibilities:
• A person with cancer tests positive • A person with cancer tests
negative
• A person without cancer tests positive • A person without cancer tests
negative

12. Suppose a test for cancer is known to be 98% accurate. This means
that the outcome of the test is correct 98% of the time. Suppose that
0.5% of the population have cancer. What is the probability that a
person who tests positive for cancer has cancer?
(1) (a)How many of the people tested have cancer?
(b) How many do not have cancer?
(2) Assume the test is 98% accurate when the result is positive.
(a) How many people with cancer will test positive?
(b) How many people with cancer will test negative?
(3) Assume the test is 98% accurate when the result is negative.
(a) How many people without cancer will test positive?
(b) How many people without cancer will test negative?
(4) (a) How many people tested positive for cancer?
(b) How many of these people have cancer?
(c) What is the probability that a person who tests positive for
cancer has cancer?

13. Homework: Exercise #41