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?Mutually Exclusive Events

and Using "And" and "Or"

Wondering by flickr user sektordua

Breakfast for RupertRupert has either milk or cocoa to drink for breakfast with either oatmeal or pancakes. If he drinks milk, then the probability that he is having pancakes with the milk is 2/3. The probability that he drinks cocoa is 1/5. If he drinks cocoa, the probability of him having pancakes is 6/7.

a) Show the sample space of probabilities using a tree diagram or any other method of your choice.

b) Find the probability that Rupert will have oatmeal with cocoa tomorrow morning.

HOMEWORK

Testing for independence ...30% of seniors get the flu every year. 50% of seniors get a flu shot annually. 10% of seniors who get the flu shot also get the flu. Are getting a flu shot and getting the flu independent events?

P(shot) = 0.50P(flu) = 0.30P(shot & flu) = (0.50)(0.30) = 0.15

HoweverP(shot & flu) = 0.10

HOMEWORK

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?

HOMEWORK

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?HOMEWORK

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.

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 ExclusiveA B1 2

3 45 6

A B

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:

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.

P(A U B) = P(A) + P(B) - P(A ∩ B) = 4/52 + 13/52 - 1/52 = 16/52

P(A and B) = P(A ∩ B) = P(A) * P(B) = (4/52) * (13/52) = 1/52

So, the probability of drawing either a king or a spade is 16/52 = 4/13.

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A B

Identify the events as:

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

d. One card - a red king or a black queen - 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.

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

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.

not mutually exclusivemutually exclusive orDropand Drag

mutually exclusive

not mutually exclusive

not mutually exclusive

mutually exclusive

not 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.

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)

A BA BA∪B A∩B

or and

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

The result is often written as follows, using set notation:

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?

HOMEWORK

The probability that Tony will move to Winnipeg is 2/9, and the probability that he will marry Angelina if he moves to Winnipeg is 9/20. The probability that he will marry Angelina if he does not move to Winnipeg is 1/20. Draw a tree diagram to show all outcomes.

1. What is the probability that Tony will move to Winnipeg and marry Angelina?

3. What is the probability that Tony does not move to Winnipeg and does not marry Angelina?

2. What is the probability that Tony does not move to Winnipeg but does marry Angelina?

HOMEWORK

(a) How many different 4 digit numbers are there in which all the digits are different?

(b) If one of these numbers is randomly selected, what is the probability it is odd?

(c) What is the probability it is divisable by 5?

HOMEWORK