Unit 7: Probability

Prof. Carolyn Dupee

July 3, 2012

7.1 EMPIRICAL PROBABILITY

•The relative frequency of occurrence of an event is determined by actual observations of an experiment

•P (E) = # of times event E has occurredtotal # of times the experiment has been performed

p. 270, Ex. 1: In 100 tosses of a fair coin, 44 landed heads up. Find the empirical probability of the coin landing heads up.

P(E) = 44 = 0.44 100

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P. 270-1, EX. 2

•A pharmaceutical company is testing a drug that is suppose to help with weight reduction. The drug is given to 500 individuals with the following outcomes. Find the empirical probability the weight is:

a) Reduced b) unchanged c) increased

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Weight reduced Weight unchanged

Weight increased

379 62 59

7.2 THEORETICAL PROBABILITY

•P(E) = # of outcomes favorable to Etotal # of possible outcomes

p. 278, Ex. 1: A die is rolled. Find the probability of rolling:a)a 3= P(3) = 1/6b)An even # P= P(2,4,6) = 3/6 or 1/2c)A # greater than 2d)A 7e)A # less than 7

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P. 281, EX. 3 SELECTING A CARD FROM DECK

•Find the probability that the one card selected is:a)a 5= P(5) = 4/52 = 1/13b)Not a 5= P(not a 5) = 1 – 1/13 OR 13/13-1/13 = 12/13c)A diamond=d)A jack or a queen or a king=e)A heart and a club=f)A card that is greater than 6 and less than 9

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SECTION 7.4 ODDS (P. 287)

•Odds- determine the probability of success and the probability of failure; compare or divide the two fraction (remember that really means you invert the second fraction and multiply!)

•Ex. 1: Determine the odds against rolling a 4 on one roll of a dice.P(4) = 1/6 P(not a 4)= 5/6P (not a 4) = 5 = 5 ÷ 1 = 5 • 6 = 5 : 1 or 5 to 1 6 6 6 6 1 P (4) 1

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DETERMINING THE PROBABILITY FROM ODDS

p. 289 Ex. 4: The odds against Robin Murphy being admitted to the college of her choice are 9:2. Determine the probability that:a)Robin is admitted (odds for)= 2/11b)Robin is not admitted (odds against)= 9/11

More practice questions: p. 274 11-15, 17; p. 283 14-16, 17-20, 24; 27-34; p. 292-293 36, 41-46, 53

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7.4 EXPECTED VALUE (EXPECTATION)E= P1•A1 + P2• A2 + P3•A3

p. 295 Ex. 1 A New Business Venture: There is a 60% change of making a $900,000 profit, a 10% chance of breaking even, and a 30% chance of losing $1,400,000. How much can JetBlue Airways “expect” to make on this new route?

Jet Blue’s expectation: P1•A1 + P2•A2 + P1•A1

Convert probabilities to decimals and multiply be each amount.(0.60)(+$900,000) + (0.10)($0) + (0.3)(-$1,400,000)

$540,000 + $0 - $420,000 =$120,000

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P. 299 EX. 7 EXPECTATION & FAIR PRICE

•You are playing a game using the spinner shown. If it costs $8 to play the game, determine the a) expectation of a person who plays the game, b) the fair price.

a)Step 1: Divide circle into smallest possible parts. Determine probabilities.$2= 2/8 = ¼ $5= 3/8 $10= 2/8= ¼ $20= 1/8Step 2: Multiply amounts times probabilities.

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$2

$5 $10

$20

$5

P. 299, EX. 7 EXPECTATION & FAIR VALUE

Amount shown on Wheel: $2 $5 $10 $20 values givenProbability ¼ 3/8 ¼ 1/8Amount won/lost -$6 -$3 $2 $12

(1/4)(-$6) + (3/8)(-$3) + (1/4)($2) + (1/8)($12) =-$1.50 - $1.125 + $0.50 + $1.50 = - $0. 625 (Expectation)Fair Price (or Value)= expectation + cost to play-$0.625 + $8 = $ 7.38

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P. 307, EX. 1

Two balls are to be selected without replacement from a bag that contains one red, one blue, one green, and one orange ball.

a)Use the counting principle to determine the number of points in the sample space.

Sample space- all the possible outcomes of the experiment 4 colors • 3 possibilities after chosen the first= 12

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P. 307, EX. 1b) Construct a tree diagram and list the sample space

Sample Space: RB, RG, RO BR, BG, BO

GR, GB, GOOR, OB, OG

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red

blue

green

orange

blue

green

orange

green

red

orangered

blue

orangeblue

green

red

P. 307

c) Determine the probability that one orange ball is selected6/12 = ½ or 1:2

d)Determine the probability that a green ball followed by a red ball is selected.

1/12 or 1:12

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7.6 OR(ADD) & AND (MULTIPLY) PROBLEMS

•P. 318 Ex. 3 Probability of A or B: One card is selected from a standard deck of playing cards. Determine whether the following pairs of events are mutually exclusive (CANNOT happen at the same time) and find P(A or B).

a)A= an ace, B= a jack (must select 2 cards)P(ace or jack)= P(ace) + P(jack)= 4 + 4 = 8 ÷ 4 = 2

52 52 52÷4 13 Mutually exclusive= an ace can’t be a jack

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P. 318, EX. 3 CONT.

•P(ace or a heart), A= an ace, B= a heart•P(ace)= 4/52•P(heart)= 13/52•P(ace and a heart)= 1/52•P(ace or heart)= 4 + 13 - 1 = 16÷4 = 4

52 52 52 52÷4 13NOT mutually exclusive because one card (1/52) can be both an ace and a heart (subtract it so it isn’t counted twice).

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P. 318, EX. 3

c) A= a red card, B= a black cardIf 2 cards are drawn, is this mutually exclusive?What is the probability of drawing a red card or a black card?

d) A= picture card, B= a red cardIf 2 cards are drawn, is this mutually exclusive?What is the probability of drawing a picture card or a red card?

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AND PROBLEMS

•P(A and B)= P(A) • P(B)•P. 320, Ex. 4: Two cards are to be selected with replacement from a deck of cards. Determine the probability that 2 queens will be selected?

4• 4 = 16 ÷16 = 152 52 2704 ÷16 169

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AND PROBLEMS

p. 320, Ex. 5: Two cards are to be selected without replacement from a deck of cards. Determine the probability that 2 queens will be selected.

P(2 queens) = P(queen 1) • P(queen 2)4• 3 = 12 ÷ 12 = 152 51 2652 ÷12 221

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INDEPENDENT & DEPENDENT EVENTS• Independent events- the occurrence of one event doesn’t affect the probability of the other event.

•Dependent events- without replacement, one event doesn’t affect the other event

•100 people attended a charity benefit to raise money for cancer research. 3 people in attendance will be selected at random without replacement, and each will be awarded one door prize. Are the events of selecting the three people who will be awarded independent or dependent events?

•Dependent b/c ea. Time one person is picked it affects results of next person getting a prize.

•P. 304-5: 54, 57, 58; p. 314-5: 27, 28; p. 327: 93-97

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