Phone NumbersA computer is used to generate random telephone numbers. Of the numbers generated and in service, 56 are unlisted and 144 are listed in the telephone directory. If one of these telephone numbers is randomly selected, what is the probability that it is unlisted?

Dependent and independent probabilities ...

A bag contains 6 marbles, 3 red and 3 blue. A marble is chosen at random and then replaced back in the bag. A second marble is selected, what is the probability that it is blue?

Independent EventsEvents in which the outcome of one event does not affect the outcome of the other event.

Dependent and independent probabilities ...

A bag contains 6 marbles, 3 red and 3 blue. A marble is chosen at random and NOT replaced back in the bag. A second marble is selected, what is the probability that it is blue?

Dependent EventsIf the outcome of one event affects the outcome of another event, then the events are said to be dependent events.

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?

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?

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

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

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.

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:

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)

Identify the events as independent, dependent, mutually exclusive, or 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.

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.

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?