Medical Testingor

Why are doctors so darn cagey?!?

Let me check your sugar. by flickr user Cataract eye

The probability that Gallant Fox will win the first race is 2/5 and that Nashau will win the second race is 1/3.

3. What is the probability that at least one horse will win a race?

2. What is the probability that both horses will lose their respective races?

1. What is the probability that both horses will win their respective races?

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

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?

Mutually Exclusive Events ...Two events are mutually exclusive (or disjoint) if it is impossible for them to occur together.

Mutually Exclusive

Formally, two events A and B are mutually exclusive if and only if

Mutually Exclusive Events ...Two events are mutually exclusive (or disjoint) if it is impossible for them to occur together.

Mutually Exclusive

Formally, two events A and B are mutually exclusive if and only if

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.

Mutually Exclusive Events ...Two events are mutually exclusive (or disjoint) if it is impossible for them to occur together.

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.

Mutually Exclusive Not Mutually Exclusive

Formally, two events A and B are mutually exclusive if and only if

ExampleSuppose 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:

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.

Probability of non-Mutually Exclusive Events ...

Not Mutually Exclusive

Identify the events as:

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.

b. One card - a red card or a king - is randomly drawn from a deck of cards.

c. A class president and a class treasurer are randomly selected from a group of 16 students.

d. One card - a red king or a black queen - is randomly drawn from a deck of cards.

e. Rolling two dice and getting an even sum or a double.

independent

dependent mutually exclusive

not mutually exclusiveDrag'n Drop

Baby!

independent

n/a

independent

dependent

n/a mutually exclusive

not mutually exclusive

mutually exclusive

not mutually exclusive

mutually exclusive

Probabilities involving "and" and "or" A.K.A "The Addition Rule"...

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:

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)

Not Mutually Exclusive

P(A or B) = P(A∪B) = P(A)+P(B) - P(A∩B)

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

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?

Homework: Exercise #41