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Chapter 3: Probability

Chapter 3: Probability. Section 3-1: Introduction. The decision to work out or foreclose a loan depends on the probability of success or failure of the workout. This will determine how much money we will recover from the loan. Definitions.

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Chapter 3: Probability

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  1. Chapter 3: Probability

  2. Section 3-1: Introduction • The decision to work out or foreclose a loan depends on the probability of success or failure of the workout. This will determine how much money we will recover from the loan.

  3. Definitions • A Trial is an activity where the result is unknown. (this is sometimes called an experiment, or a random experiment) • An Outcome is one specific result of a trial. • A Sample Space is the set of all possible outcomes. It is usually represented by a capital S, but we will use the symbol § to represent sample space. (S is needed for something else) • A Probability is the proportion of times a particular event will occur. It is usually represented by a capital P.

  4. Flip a coin 10 times. • Record the number of times a “head” comes up. • Add up the total number for the whole class • What do you observe? • How many heads out of number of flips?

  5. Flipping a coin is a trial • A “H” or “T” are outcomes of the trial. • Sample Space = {H, T} • P(H) = ½ • P(T) = ½

  6. Example • You have three nickels (coins) that you flip into the air and onto the table. • A trial would be flipping the three coins. • One outcome would be heads-tails-heads (HTH for short). What are the other possible outcomes? (see next slide) • An event is a subset of outcomes from a sample space

  7. Example • The samplespace for this trial of flipping 3 coins (8 in this case.) • HHH HHT • HTH HTT • TTT TTH • THT THH

  8. Some Notation • Suppose we are interested in knowing all the possible outcomes where we get two heads and one tail. We will call this event A: • We can write A in set notation: • A = {HHT, HTH, THH} • There are therefore 3 ways this can happen out of a total of 8 possible outcomes. We say the probability of the event A happening is 3/8.

  9. For example, when rolling one die, § = {1,2,3,4,5,6}. • Let the event E = rolling an even number. Then the set E is all the ways to roll an even number: E = {2,4,6}. This is a subset of §.

  10. Theoretical Probability • Let E be an event. Then: • P(E) = k/n • k = the number of ways event E can occur • n = the total number of possible outcomes • *CAUTION: This formula is only valid if each outcome is equally likely.

  11. Example • A marble is drawn from a bag. There are 15 red, 12 yellow, and 18 blue marbles in the bag. • What is the probability of randomly drawing a single red marble from the bag? • What is the probability of randomly drawing a single blue marble from the bag?

  12. Question for Discussion (something to think about. But we will discuss later) • A marble is drawn from a bag. There are 15 red, 12 yellow, and 18 blue marbles in the bag. • What is the probability of drawing a red marble from the bag, setting it aside, and then immediately drawing a second red marble? • Does the result change if you replace the first red marble before drawing the second time? If so, why?

  13. Empirical Probability • Empirical data is that which you observe. For example, you have collected data that indicates that of the last 550 loans a bank granted, 42 of them were foreclosed upon. • Then based on the empirical data, you might say that the probability of a loan going into foreclosure is 42/550.

  14. Empirical Probability • Let E be an event. Then: • P(E) = k/n • k = the number of times event E has occurred inthepast (under similar circumstances) • n = the number of trials inthepast

  15. Law of Large Numbers • Empirical probabilities are basically just estimates. They do not necessarily predict the outcome of a particular trial. (“The next loan will go to foreclosure!”) We do know this, however: The outcome of one trial cannot be predicted, but, one can predict what will happen over a series of many trials.

  16. 3-2: Combining Events • A marble is drawn from a bag. There are 15 red, 12 yellow, and 18 blue marbles in the bag. • What is the probability of randomly drawing either a yellow or blue marble from the bag? • What about: “What is the probability of not picking a red marble?” • How are these related to the topics we discussed in Chapter 2?

  17. 3-3: Dice Problem • Suppose you have two dice that you roll onto a table. Here is the sample space.

  18. Questions • What is the probability of getting a total of 10? • What is the probability of getting the same number on each die? • What is the probability of getting a prime number total? • What is the probability of NOT getting a total of 7? • What is the probability of getting either a total of 8 or a total of 11? • What is the probability of getting either a 4 or an odd number on one of the dies? • What is the probability of getting a 4 and an odd number on the other die?

  19. Rolling a die • Let event A = {an even number} • Event B = {an odd number} • Note that A and B are disjoint • We call these events mutually exclusive events when they don’t have any outcomes in common.

  20. 3-5: Rules of Probability • Let A and B be events, and let § be the sample space. • Rule 1: 0 P(A)  1 • Rule 2: P(§) = 1 • Rule 3: P(AC) = 1 – P(A) • Rule 4: If A and B are mutually exclusive events, then • P(A B) = P(A) + P(B)

  21. Mutually exclusive versus Independent events: When you roll a die, the events { odd} and {even} are mutually exclusive because they cannot happen at the same time. Hence, P(odd or even) = P(odd) +P(even) When you roll a die two times, the events that the 1st die is odd and the 2nd die is odd are independent events. The events does not affect or influence each other. Hence, Probability = P(odd) *P(odd) = 3/6*3/6

  22. Not mutually exclusive events Roll a die….let us define C = number that comes up is 1 or 2 or 3 D = number that comes up is 3 or 4 or 5 Hence, C and D are not mutually exclusive P(C or D) = P(C) + P(D) - P(C and D) = 3/6+3/6 - 1/6 = 5/6

  23. Rolling two dice: • What is the probability of getting a “4” and an odd number on the other die? Independent events: hence, 1/6 * 3/6 = 3/36 {(4,1) (4,3), (4,5) }

  24. A marble is drawn from a bag. There are 15 red, 12 yellow, and 18 blue marbles in the bag. • What is the probability of drawing a red marble from the bag, setting it aside, and then immediately drawing a second red marble? (not independent events) • Does the result change if you replace the first red marble before drawing the second time? If so, why? (independent events)

  25. (a) 15/45 * 14/44 (b) 15/45 * 15/45 NOTE: Probability that the 2nd marble is red knowing that the 1st marble is red and it is not replaced is just 14/44

  26. 3-6: Venn Diagrams • A total of 70 students are randomly interviewed. 23 own a car. 45 own a bike. 18 own both a car and a bike. Draw a Venn diagram that displays all of the probabilities related to this survey.

  27. Find the probabilities: • A student randomly chosen owns a car but not a bike. • A student randomly chosen does not own either a bike or a car. • A student randomly chosen owns a car or a bike. • A student randomly chosen owns a car and a bike.

  28. 3-6B General Probability Formula • If A and B are events, then • P(AB) = P (A) + P (B) – P (AB) • Subtracting compensates for the double-counting error.

  29. Suppose 8% of a certain batch of calculators have a defective case, and that 11% have defective batteries. Also, 3% have both a defective case and defective batteries. A calculator is selected from the batch at random. Find the probability that the calculator has a good case and good batteries.

  30. Ms Bezzone invites 10 relatives to a party: her mother, 2 uncles, 3 brothers, and 4 cousins. If the chances of any one guest arriving first are equally likely, find the probabilities: • The first guest is an uncle or a cousin. • The first guest is a brother or a cousin. • The first guest is an uncle or her mother.

  31. The table shows the probability of a person accumulating credit card charges over a 12-month period:

  32. Find the probability that a person’s total charges during the period are • $500 or more • Less than $1000 • $500 to $2999 • $3000 or more

  33. Questions • For the following problems, the trial is rolling two dice ( a red and a green die) (be sure to avoid double-counting) • What is the probability of sum of both dice being 7? • What is the probability that the red die will show an odd number or the sum of the two dice will be 8? • What is the probability that the green die is 6 or the sum of the two dice is 10? • What is the probability that the green die shows an even number and the sum of the two dice is 10?

  34. Group Exercise • See #36 from Chapter 3 • From a survey involving 1,000 people in a certain city, it was found that 500 people had tried a certain brand of diet cola, 600 had tried a certain brand of regular cola, and 200 had tried both types of cola. (Barnett p. 414)

  35. Group Exercise • Draw and label a Venn Diagram that demonstrates this information in sets. Then try another Venn diagram using the probabilities. • What is your sample space? • Find the probability that a randomly selected person from the city has tried both of the colas. Find the probability that a randomly selected person from the city has tried the diet cola but not the regular cola. • Find the probability that a randomly selected person from the city has tried the regular cola but not the diet cola.

  36. Group Exercise • Find the probability that a randomly selected person from the city has tried neither of the colas. • Find the probability that a randomly selected person from the city has tried either the diet or the regular cola. Try to see if you can compute this in TWO different ways. • Find the probability that a randomly selected person from the city has tried one of the colas but not both. Write your answer in probability notation. Be careful on this one…a picture should help.

  37. Focus on the Project • Let S be the event that an attempted work out is successful and let F be the event that it fails. Use the COUNTIF function to find the fraction of past work outs which were successful. This fraction is our estimate for P(S). Likewise, we find the fraction of attempts that failed and use this as our estimate for P(F).

  38. Focus on the Project • Example 1: • Use the Loan Records.xls to estimate P(S). • DCOUNT will work • COUNTIF will also do the job

  39. Focus on the Project • Go to the section titled Project 1 Specifics (Chapter 7) and do Part 2a only. • Also do Chapter 3 Focus On Project Memo. • Edit your Written Report to reflect your newest information. Be sure to use proper probability and mathematics notation in your writing (use the Equation Editor to format all mathematical text).

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