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Unformatted text preview: Statistics 10: Week Five Session Two (Chapter 14)
Professor Esfandiari Major Concepts: Trial: An observation made on a random phenomenon
Outcome: The value assigned to the random phenomenon
Event: Combination of outcomes Sample space: Collection of all possible outcomes Example: Suppose you are give a multiple choice test with three items on a language you
have never studied, heard, or spoken and you are choosing the correct answer at random.
Each question has four options only one of which is correct. Trial: The random phenomenon is choosing the correct options at random
Outcome: Is the probability of M1 assigned to picking the right answer at random and 3/1 assigned to the wrong answer.
Event: is the combination of outcomes. For instance getting two right and one wrong answer can happen three ways (see below)
Sample space: is the collection of all possible outcomes including Example of sample space of possible collection of outcomes: Three correct answers Right Right Right P: 1A’kl/4i‘l/4 = 1/64
One possible outcome Two correct answers Right Right Wrong P = l11’1‘1/4’k3/4 = 3/64 Three possible outcomes Right Wrong Right 3*3/64 = 9/64
Wrong Right Right One correct answer Wrong Wrong Right P = 3/:’*‘_°)/4"‘l/4 = 9/64 Three possible outcomes Wrong Right Wrong 3*9/64 = 27/64
Right Wrong Wrong No right answers
One possible outcome Wrong Wrong Wrong 3/1>“3/4*3/4 = 27/64 Total number of outcomes = 8.
Total probability for the sample space = 1/64+9/64+27/64+27/64 = 64/64 = 1 Law of Large Numbers (LLN) As the number of independent trials increases, the longterm relative frequency of the
repeated event becomes closer to the probability of that event. In the above example, choosing of correct answers at random is an independent event and the
chance of getting any item correct is equal to 0.25; the same way that if we repeatedly toss a coin
the probability of getting a head on any of the events stays constant and it equal to 0.50. Relative frequency: Number of times an event happens/total of trials Example: If you toss a coin 100 times and get 45 heads Relative frequency: 45/ 100 Empirical probability: long run relative frequency of the occurrence of an event Theoretical probability: comes from a model Example: Probabilities resulting from the N ormal model is an example of theoretical
probability ° If we pick a person at random the probability that his/her score be within one
standard deviation of the mean is 68%. ' If we pick a person at random the probability that his/her score be within two
standard deviations of the mean is 95%. ° If we pick a person at random the probability that his/her score be within three
standard deviations of the mean is 99.7%. Probability Of an outcome = Number of time the outcome happens/Total number of outcomes Example: P (getting three right answers in the above example) 2 1/8 Some Probability Rules Probability of any event is between zero and one
0 < P (any event) < :1 Probability for set of all outcomes = 1
P (S) = 1 Complement: The set of outcomes that are not part of the event of our interest Example: Event of interest (getting all three answers correct
P (right answers = 3) = 1A*1/4*1/4 = 1/64
Probability of the complement of three correct answers P (two or one or no correct answers) 2 l — 1/64 = 64/64 — 1/64 2 63/64
Or (9/64+ 27/64 + 27/64) = 63/64 Disjoint events: Do not overlap. When one happens, the other once cannot happen. They
are also called mutually exclusive. They are not independent. Example: One cannot have blood type A and O at the same time. These events do not
overlap. They are disjoint. Addition rule for disjoint events:
P(A or B) = P(A) + P(B) Independent events: The outcome of one event does not inﬂuence the outcome of the
other. Example: In the example given above if you randomly choose the right answer for the
first question, it has no inﬂuence on randomly choosing the right answer for the second
question, and if you randomly choose the correct answer for the first and the second
question, it does not stop you from randomly choosing the right answer for the third
questlon. Multiplication rule for independent events:
P (A and B) = P(A)* P(B) This mle can be generalized to more events P (A and B and C) = P(A)* P(B)*P(C) P (randomly picking the right answer for the first, second, and third question)
= 1/4* 1/4* 1/4 : 1/64 Question to be solved in class: Supp05e that you were a teaching assistant and you had decided to give a quiz with four
questions to check on your students’ understanding of the material. a) If you wanted to minimize the chance of choosing the correct answer at random,
would you go for True/False or multiplechoice version? Why? Assume that the
multiple—choice version has four options for each question only one of which is
correct. b) What would be the probability of randomly getting all of the four correct answers
be in the multiple choice version? c) What would be the probability of randomly getting all of the four correct answers
be in the True/False version? d) Draw a sample space for possible outcomes for a quiz with four questions and
compute the relevant probabilities for True/False and Multiple/Choice option. For each of the following, list the 'sam— Ce and tell whether you think the events are
likely: airplanesliave an excellent safety
there are crashes occasionally.with:., 8. Crash. Commercial
record. Nevertheless, the loss of many lives. In theweeks follong a crash, air lines often report axdrop in the number of passengers, probably because people are afraid to risk ﬂying. a) A travel agent Suggests that, since the law of averages
makes it highly unlikely to have two plane crashes . within a few Weeks of each other, ﬂying soon after a crash is the safest time. What do you think? b) If the airline industry proudly announces that it has
set a new record for the longest period of safe ﬂights, would you be reluctant to ﬂy? Are the airlines due to
have a crash? Avuuvamlé ; 20, Stats projects. In a large Introductory Statistics lecture
hall, the professor reports that 55% of the students en
rolled have never taken a Calculus course, 32% have
taken only one semester of Calculus, and the rest have
taken two or more semesters of Calculus. The professor
randomly assigns students to groups of three to work on
a project for the course. What is the probability that the
ﬁrst groupmate you meet has studied a) two or more semesters of Calculus? b) some Calculus? c) no more than one semester of Calculus? g” 31. M&M’ . The Masterfoods company says that before the
introduction of purple, yellow candies made up 20% of
their plain M&M’s, red another 20%, and orange, blue,
and green each made up 10%. The rest were brown. a) If you pick an M&M at random, What is the probabil
ity that
1) it is brown?
2) itis yellow or orange?
'3) not green?
4) it is striped?
b) If you pick three M&M’s in a row,
bility that
1) they are all brown?
2) the third one is the first one that’ 3 red?
3) none are yellow?
4) at least one is green? . Blood. The American Red Cross says that about 45% of
the US. population has Type 0 blood, 40% Type A, 11% \
Type B, and the rest Type AB. , a) Someone volunteers to give blood. What is the
probability that this donor
1) has Type AB blood?
2) has Type A or Type B?
3) is not Type O?
b) Among four potential donors,
that
1) all are Type O?
2) no one is Type AB?
3) they are not all Type A?
4) at least one person is Type B? what is the proba— what is the probability 33. Disjoint or independent? In Exercise 31 you calcu
lated probabilities of getting various M&M’s. Some of
your answers depended on the assumption that the out . comes described Were disjoint; that is, they could not both ’ happen at the same time. Other answers depended on the
assumption that the events were independent; that is, the
occurrence of one of them doesn’t affect the probability of
the other. Do you understand the difference between dis—.
joint and independent? a) If you draw one M&M, are the events of getting a red one and getting an orange one disjoint, inde—
pendent, or neither? ...
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 Fall '11
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