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Part 3 Module 3 Probability

To introduce the classical definition of probability , here is a question that you can probably answer, intuitively, regardless of whether or not you have studied probability. Suppose we roll one die. Q: What is the probability that the result of the die roll will be a “two”?

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Part 3 Module 3 Probability

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  1. To introduce the classical definition of probability, here is a question that you can probably answer, intuitively, regardless of whether or not you have studied probability. Suppose we roll one die. Q: What is the probability that the result of the die roll will be a “two”? A: We would probably say that the probability is “one out of six,” or something equivalent to that. This is because we know that when we roll one die, there are six equally likely outcomes. Of the six equally likely outcomes, one of them is a “two.” Part 3 Module 3 Probability

  2. In this course, we will express probabilities as numbers, rather than as ratios. Instead of saying that the probability of rolling a two is “one out of six,” we will say that the probability is 1/6 or .167 (the decimal equivalent of 1/6, rounded) Calculating a probability is equivalent to calculating a percent. Probabilities are numbers

  3. A random experiment is any process whose outcome isn’t known in advance. Rolling a die is an example of a random experiment. Terminology: random experiment

  4. The sample space(S) for a random experiment is the set of all equally-likely outcomes. For the experiment of rolling one die, the sample space is S = {1, 2, 3, 4, 5, 6} Terminology: sample space

  5. An event (E) is any outcome or combination of outcomes in a random experiment. An event is always a subset of the sample space for the random experiment. In the previous example, the experiment consisted of rolling one die, and the event (E) we were interested in was “the result of the die roll is a two.” E = {2} Terminology: event

  6. Using the terminology just introduced, we have the following classical or theoretical definition of the probability of an event E. The classical definition of probability

  7. Probability In much of our work in Part 3 Modules 3 through 5, we will work with experiments that involve selecting or more individuals from a specified population. The simplest situation is one in which a single individual is being selected. At the Wee Folks Gathering there are 5 elves (E) 2 hobbits (H) 8 gnomes (G) They will randomly select one person from amongst this group to serve as Grand Marshall. 1. What is the probability that the randomly selected person is a gnome (G)? 2. What is the probability that the randomly selected person is an elf (E)? Answers P(G) = 8/15 = .5333 P(E) = 5/15 = .3333

  8. Probability At the Wee Folks Gathering there are 5 elves (E) 2 hobbits (H) 8 gnomes (G) They will randomly select one person from amongst this group to serve as Grand Marshall. 1. What is the probability that the randomly selected person is not a gnome (G´)? 2. What is the probability that the randomly selected person is not an elf (E´)? Answers P(G´) = 7/15 = .4667 P(E´) = 10/15 = .6667

  9. Let E be any event. The complement of E, denoted E´ is the non-occurrence of E, or the opposite of E. In the previous examples, for instance, note that the probability of selecting a gnome was .5333 [that is, P(G) = .5333] and the probability of not selecting a gnome was .4667 [that is, P(G´) = .4667] Also note that these two probabilities have a special relationship: P(G) + P(G´) = .5333 + .4667 = 1 This illustrates an important fact called the Complements Rule. Complements

  10. The Complements Rule For any event E in any experiment, P(E) + P(E´) = 1 This rule is usually presented in a slightly different, but equivalent, form: The Complements Rule For any event E in any experiment, P(E´) = 1 – P(E) The Complements Rule

  11. Among a certain group of Vikings, 28 of them like to pillage, 18 of them like to plunder, while 10 of them like to pillage and like to plunder and 12 of them don't like to pillage and don't like to plunder. If one of these Vikings is randomly selected, find the probability that he/she likes to pillage.
 A. 0.4828 B. 0.3103 C. 0.5833 D. 0.2800 A different idea

  12. Among a certain group of Vikings, 28 of them like to pillage, 18 of them like to plunder, while 10 of them like to pillage and like to plunder and 12 of them don't like to pillage and don't like to plunder. If one of these Vikings is randomly selected, find the probability that he/she likes to pillage.
A. 0.4828 B. 0.3103 C. 0.5833 D. 0.2800 Because these categories (pillage, plunder) are not mutually exclusive, a Venn diagram will be helpful. The diagram shows that the total population is 48, and 28 of them like to pillage, so the probability is 28/48 = .5833 Solution

  13. Among a certain group of Vikings, 28 of them like to pillage, 18 of them like to plunder, while 10 of them like to pillage and like to plunder and 12 of them don't like to pillage and don't like to plunder. If one of these Vikings is randomly selected, find the probability that he/she doesn’t like to pillage.
 Solution: (We will refer to the answer to the previous question, and use the complements rule) P(doesn’t like to pillage) = 1 – P(likes to pillage) = 1 – .5833 = .4167 Another exercise

  14. Two events are said to be mutually exclusive if it is impossible for the two events to occur simultaneously. If E, F are mutually exclusive events, then P(E or F) = P(E) + P(F) Mutually exclusive events

  15. At the Wee Folks Gathering there are 5 elves (E) 2 hobbits (H) 8 gnomes (G) They will randomly select one person from amongst this group to serve as Grand Marshall. What is the probability that the randomly selected person is a gnome or an elf? Note that these events (G, E) are mutually exclusive: it is possible that the selected person might be a gnome, and it is possible that the selected person might be an elf, but it is impossible for the selected person to be both a gnome and an elf. P(G or E) = P(G) + P(E) = .5333 + .3333 = .8666 Mutually exclusive events

  16. Statistical Probability Certain statistics are also probabilities. In this course, we will work frequently with population statistics. Statistical probability is inferred from population statistics.

  17. Statistical Probability A few years ago, for example, the Natural Resources Defense Council conducted a study of bottled water (this example is non-fiction). They found that 40% of bottled water samples were merely tap water. They also found that 30% of bottled water samples were contaminated by substances such as arsenic and fecal bacteria.

  18. Statistical Probability 40% of bottled water samples are merely tap water. 30% of bottled water samples are contaminated by substances such as arsenic and fecal bacteria. These unappetizing statistics are probabilities. Let T be the event that a randomly selected sample of bottled water is tap water. Let C be the event that a randomly selected sample of bottled water is contaminated. Then P(T) = 40% = .4 P(C) = 30% = .3

  19. 40% of bottled water samples are merely tap water. 30% of bottled water samples are contaminated by substances such as arsenic and fecal bacteria. Let T be the event that a randomly selected sample of bottled water is tap water. Let C be the event that a randomly selected sample of bottled water is contaminated. 1. What is the probability that a randomly selected sample of bottled water is not merely tap water? 2. What is the probability that a randomly selected sample of bottled water is not contaminated? Answers: 1. P(T´) = 1 – P(T) = 1 – .4 = .6 2. P(C´) = 1 – P(C) = 1 – .3 = .7 The Complements Rule, again

  20. The table below shows the distribution of scores on Test 1 in Partial Differential Equations for Liberal Arts. Score percent 0 - 49 2% 50 - 59 6% 60 - 64 6% 65 - 69 14% 70 - 79 37% 80 - 89 17% 90 -100 18% If one student is randomly selected, what is the probability that his/her test score is in the 50-59 range or the 70-79 range? A. .43 B. .97 C. .63 D. None of these Mutually exclusive events

  21. The table below shows the distribution of scores on Test 1 in Partial Differential Equations for Liberal Arts. Score percent 0 - 49 2% 50 - 59 6% 60 - 64 6% 65 - 69 14% 70 - 79 37% 80 - 89 17% 90 -100 18% If one student is randomly selected, what is the probability that his/her test score is in the 50-59 range or the 70-79 range? These are mutually exclusive events. The probability that a student’s score in in the 50-59 range is 6% or .06; the probability that a student’s score is in the 70-79 range is 37% or .37; so, the probability that a student’s score is in the 50-59 range or the 70-79 range is .06 + .37 = .43 Solution

  22. Odds, like probability, use ratios or fractions to indicate likelihood. Odds are not the same as probability, however. According to the classical definition, that the probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes. The odds in favor of an event is the ratio of favorable outcomes to unfavorable outcomes. The odds against an event is the ratio of unfavorable outcomes to favorable outcomes. Side note: “Odds”

  23. The odds in favor of an event is the ratio of favorable outcomes to unfavorable outcomes. Refer to the experiment of rolling one die. Let E be the event that the result of the die roll is “2.” We know that P(E) = 1/6. The odds in favor of E are not 1/6, however. In this experiment, one of the six outcomes is favorable to E, and the other five outcomes are unfavorable. The odds in favor of E are 1/5, or “1 to 5.” This is also expressed as 1:5. The odds against E are “5 to 1”, or 5:1. Odds vs. probability

  24. The table below shows the distribution of scores on Test 1 in Partial Differential Equations for Liberal Arts. Score percent 0 - 49 2% 50 - 59 6% 60 - 64 6% 65 - 69 14% 70 - 79 37% 80 - 89 17% 90 -100 18% If one student is randomly selected, find the ODDS AGAINST the event that he or she has a test score in the 70 - 79 range?
 A. 63:37 B. 37:100 C. 37:63 D. 63:10 Exercise

  25. Score percent 70 - 79 37% If one student is randomly selected, find the ODDS AGAINST the event that he or she has a test score in the 70 - 79 range? The odds against an event are the ratio of unfavorable outcomes to favorable outcomes. Since 37% of the students have scores in the 70-79 range, we know that 37(%) are favorable to that event, and 63(%) are unfavorable, so the ratio of unfavorable to favorable outcomes is 63:37 (choice A).
 A. 63:37 Solution

  26. One last note: When the term “odds” is used in the context of sports betting, it refers to the odds against an event. For instance, if we are told that the odds for a certain horse winning a race are 7 to 1, that means that the odds against the horse winning are 7 to 1; the odds in favor are 1 to 7. Odds in Sports Betting

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