MA 485/585 Probability Theory (Dr Chernov)

Five cards Five cards are labeled 1,2,3,4,5. They are shuffled and lined up in an arbitrary order. How many ways can this be done? What is the chance that they are lined up in the right order: 1,2,3,4,5?

Committee A committee of 10 members decides to choose a chair and a secretary arbitrarily (at random). What is the chance that the tallest member becomes the chair and shortest – the secretary? Chair Secretary

2 Aces A deck of 10 cards contains two aces. We pick two cards arbitrarily (at random). What is the chance that both are aces?

Pascal’s triangle

Coin tossing A coin is tossed 100 times. What is the chance one observes exactly 50 Heads and 50 Tails?

10 men and 10 women A company employs 10 men and 10 women. It forms a team of three employees for a project by picking the employees at random. What is the chance that all members of the team are women?

2 dice Two dice are rolled. What is the chance that the sum of the numbers shown equals 9?

Die unfolded Sides of a die are marked by numbers: 1,2,3,4,5,6 When you roll a die, one of these numbers comes up

Two dice rolled: When two dice are rolled, two numbers come up.

Urn with black and white balls An urn contains 10 white balls and 20 black balls. Four balls are taken from the urn at random. What is the probability that two white and two black balls are taken?

Concepts of Probabilities  Event any collection of outcomes of a procedure  Outcome an outcome or a possible event that cannot be further broken down into simpler components  Probability Space (for a procedure) the collection of all possible outcomes

Probability Limits  The probability of an event that is certain to occur is 1.  The probability of an impossible event is 0.  For any event A, the probability of A is between 0 and 1 inclusive. That is, 0  P(A)  1.

Possible Values for Probabilities

Complementary Events The complement of event A, denoted by A c, consists of all outcomes in which the event A does not occur.

Venn Diagram for the Complement of Event A A is yellow, A c is pink

Rules for Complementary Events P(A) + P(A c ) = 1 = 1 – P(A) P(A) = 1 – P(A c ) P(Ac)P(Ac)

Illustration (Venn Diagram) A is the red disk, B is the yellow disk A or B is the total area covered by both disks.

Disjoint Events Events A and B are disjoint (or mutually exclusive) if they cannot occur at the same time. (That is, disjoint events do not overlap.) Venn Diagram for Disjoint Events

Addition Rule for disjoint events P(A or B) = P(A) + P(B) (only if A and B are disjoint)

Venn’s diagram for two events

Venn’s diagram for three events

Binomial probability function

Binomial probability functions for n=100 and p = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9

The Rule