Mathematics Department Probability Ernesto A. Diaz Faculty Mathematics Department 1

Theoretical Probability Concepts

Definitions An experiment is a process by which an observation or outcome is obtained The possible results of an experiment are called its outcomes. Sample Space is the set S of all possible outcomes An event is a subset E of the sample space S.

Example A dice. Events and Outcomes are not the same E1={ 3 } “A three comes up” E1 ={ 2, 4, 6 } “an even number” Events and Outcomes are not the same An event is a subset of the sample space An outcome is an element of the sample space

Definitions continued Theoretical probability (a priori) based on deductive thinking. It is determined through a study of the possible outcome that can occur for the given experiment. Empirical probability (a posteriori) based on inductive thinking. It is the relative frequency of occurrence of an event and is determined by actual observations of an experiment. Subjective It is based on individual experience

Theoretical Probability If each outcome of an experiment has the same chance of occurring as any other outcome, they are said to be equally likely outcomes.

Example A die is rolled. Find the probability of rolling a) a 3. b) an odd number c) a number less than 4 d) a 8. e) a number less than 9.

Solutions a) b) There are three ways an odd number can occur 1, 3 or 5. c) Three numbers are less than 4.

Solutions: continued d) There are no outcomes that will result in an 8. e) All outcomes are less than 10. The event must occur and the probability is 1.

Empirical Probability Example: In 100 tosses of a fair die, 19 landed showing a 3. Find the empirical probability of the die landing showing a 3. Let E be the event of the die landing showing a 3.

The Law of Large Numbers The law of large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events or precise totals.

Important Facts The probability of an event that cannot occur is 0. The probability of an event that must occur is 1. Every probability is a number between 0 and 1 inclusive; that is, 0  P(E) 1. The sum of the probabilities of all possible outcomes of an experiment is 1.

Example A standard deck of cards is well shuffled. Find the probability that the card is selected. a) a 10. b) not a 10. c) a heart. d) a ace, one or 2. e) diamond and spade. f) a card greater than 4 and less than 7.

Example continued a) a 10 There are four 10’s in a deck of 52 cards. b) not a 10

Example continued c) a heart There are 13 hearts in the deck. d) an ace, 1 or 2 There are 4 aces, 4 ones and 4 twos, or a total of 12 cards.

Example continued d) diamond and spade The word and means both events must occur. This is not possible. e) a card greater than 4 and less than 7 The cards greater than 4 and less than 7 are 5’s, and 6’s.

12.6 Or and And Problems

Or Problems P(A or B) = P(A) + P(B)  P(A and B) Example: Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a bowl and one is randomly selected. Find the probability that the piece of paper selected contains an even number or a number greater than 5.

Solution P(A or B) = P(A) + P(B)  P(A and B) Thus, the probability of selecting an even number or a number greater than 5 is 7/10.

Example Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a bowl and one is randomly selected. Find the probability that the piece of paper selected contains a number less than 3 or a number greater than 7.

Solution There are no numbers that are both less than 3 and greater than 7. Therefore,

Mutually Exclusive Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously.

Example One card is selected from a standard deck of playing cards. Determine the probability of the following events. a) selecting a 3 or a jack b) selecting a jack or a heart c) selecting a picture card or a red card d) selecting a red card or a black card

Solutions a) 3 or a jack b) jack or a heart

Solutions continued c) picture card or red card d) red card or black card

And Problems P(A and B) = P(A) • P(B) Example: Two cards are to be selected with replacement from a deck of cards. Find the probability that two red cards will be selected.

Example Two cards are to be selected without replacement from a deck of cards. Find the probability that two red cards will be selected.

Independent Events Event A and Event B are independent events if the occurrence of either event in no way affects the probability of the occurrence of the other event. Experiments done with replacement will result in independent events, and those done without replacement will result in dependent events.