1 Probability Ernesto A. Diaz Faculty Mathematics Department

2 Specific Deductive vs. Inductive Inductive Reasoning General Specific Conclusion is guaranteed Deductive Reasoning General Conclusion is probable Not always can be proved

3 Intro to Probabilities Year - AuthorExample of work : Gerolamo Cardano : Antoine Gombauld / Pascal / Fermat : John Graunt : Jacob Bernoulli : Marquis de Laplace : Gregor Mendel - Study of Probability and Gambling - Probability theory - Observations on the Bills of Death - “The art of guessing” applications on government, economics, law, genetics - Analytic Theory of Probabilities. Interpreting scientific data - Foundation of Genetics

4 Theoretical Probability Concepts

5 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.

6 Example A dice. E 1 ={ 3 } “A three comes up” E 1 ={ 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

7 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

8 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.

9 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.

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

11 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.

12 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.

13 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.

14 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.

15 XIX Century: Opposition & Synthesis Adolphe QueteletJames Clerk Maxwell - Statistics in Social Science - Work: - Patterns in human traits (e.g. height) follow normal curve - Social Statistics have similarities, e.g. rate of murder vs. suicide in Belgium - Statistics in Mechanics of Gases - Work: - Patterns in molecule behavior follow statistical trends

16 XIX Century: Opposition & Synthesis Adolphe QueteletJames Clerk Maxwell - Statistics in Social Science - Work: - Patterns in human traits (e.g. height) follow normal curve - Social Statistics have similarities, e.g. rate of murder vs. suicide in Belgium - Conclusions: - Constant social causes dictate behavior; are individuals free? - Laplace: with sufficient knowledge, nothing is uncertain - Statistics in Mechanics of Gases - Work: - Patterns in molecule behavior follow statistical trends - Conclusions: - Statistical regularities in the large scale say nothing of the behavior of individual in the small scale

17 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.

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

19 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.

20 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.

Copyright © 2005 Pearson Education, Inc Expected Value (Expectation)

Slide Copyright © 2005 Pearson Education, Inc. Expected Value The symbol P 1 represents the probability that the first event will occur, and A 1 represents the net amount won or lost if the first event occurs.

Slide Copyright © 2005 Pearson Education, Inc. Example Teresa is taking a multiple-choice test in which there are four possible answers for each question. The instructor indicated that she will be awarded 3 points for each correct answer and she will lose 1 point for each incorrect answer and no points will be awarded or subtracted for answers left blank.  If Teresa does not know the correct answer to a question, is it to her advantage or disadvantage to guess?  If she can eliminate one of the possible choices, is it to her advantage or disadvantage to guess at the answer?

Slide Copyright © 2005 Pearson Education, Inc. Solution Expected value if Teresa guesses.

Slide Copyright © 2005 Pearson Education, Inc. Solution continued—eliminate a choice

Slide Copyright © 2005 Pearson Education, Inc. Example: Winning a Prize When Calvin Winters attends a tree farm event, he is given a free ticket for the $75 door prize. A total of 150 tickets will be given out. Determine his expectation of winning the door prize.

Slide Copyright © 2005 Pearson Education, Inc. Example When Calvin Winters attends a tree farm event, he is given the opportunity to purchase a ticket for the $75 door prize. The cost of the ticket is $3, and 150 tickets will be sold. Determine Calvin’s expectation if he purchases one ticket.

Slide Copyright © 2005 Pearson Education, Inc. Solution Calvin’s expectation is  $2.49 when he purchases one ticket.

Slide Copyright © 2005 Pearson Education, Inc. Fair Price Fair price = expected value + cost to play

Slide Copyright © 2005 Pearson Education, Inc. Example Suppose you are playing a game in which you spin the pointer shown in the figure, and you are awarded the amount shown under the pointer. If is costs $10 to play the game, determine a) the expectation of the person who plays the game. b) the fair price to play the game. $10 $2 $20$15

Slide Copyright © 2005 Pearson Education, Inc. Solution $0 3/8 $10 $5  $8 Amt. Won/Lost 1/8 3/8Probability $20$15$2Amt. Shown on Wheel

Slide Copyright © 2005 Pearson Education, Inc. Solution Fair price = expectation + cost to play =  $ $10 = $8.87 Thus, the fair price is about $8.87.

Copyright © 2005 Pearson Education, Inc Or and And Problems

Slide Copyright © 2005 Pearson Education, Inc. 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.

Slide Copyright © 2005 Pearson Education, Inc. 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.

Slide Copyright © 2005 Pearson Education, Inc. 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.

Slide Copyright © 2005 Pearson Education, Inc. Solution There are no numbers that are both less than 3 and greater than 7. Therefore,

Slide Copyright © 2005 Pearson Education, Inc. Mutually Exclusive Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously.

Slide Copyright © 2005 Pearson Education, Inc. 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

Slide Copyright © 2005 Pearson Education, Inc. Solutions a) 3 or a jack b) jack or a heart

Slide Copyright © 2005 Pearson Education, Inc. Solutions continued c) picture card or red card d) red card or black card

Slide Copyright © 2005 Pearson Education, Inc. 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.

Slide Copyright © 2005 Pearson Education, Inc. Example Two cards are to be selected without replacement from a deck of cards. Find the probability that two red cards will be selected.

Slide Copyright © 2005 Pearson Education, Inc. 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.

Slide Copyright © 2005 Pearson Education, Inc. Example A package of 30 tulip bulbs contains 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. Three bulbs are randomly selected and planted. Find the probability of each of the following.  All three bulbs will produce pink flowers.  The first bulb selected will produce a red flower, the second will produce a yellow flower and the third will produce a red flower.  None of the bulbs will produce a yellow flower.  At least one will produce yellow flowers.

Slide Copyright © 2005 Pearson Education, Inc. Solution 30 tulip bulbs, 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. All three bulbs will produce pink flowers.

Slide Copyright © 2005 Pearson Education, Inc. Solution 30 tulip bulbs, 14 bulbs for red flowers, 0010 for yellow flowers, and 6 for pink flowers. The first bulb selected will produce a red flower, the second will produce a yellow flower and the third will produce a red flower.

Slide Copyright © 2005 Pearson Education, Inc. Solution 30 tulip bulbs, 14 bulbs for red flowers, 0010 for yellow flowers, and 6 for pink flowers. None of the bulbs will produce a yellow flower.

Slide Copyright © 2005 Pearson Education, Inc. Solution  30 tulip bulbs, 14 bulbs for red flowers, 0010 for yellow flowers, and 6 for pink flowers. At least one will produce yellow flowers. P(at least one yellow) = 1  P(no yellow)