Unit 7: Probability

7.1: Terminology I’m going to roll a six-sided die. Rolling a die is called an “experiment” The number I roll is called an “outcome” An “event” is a group of outcomes Example of event: roll an even number This event includes the outcomes 2, 4, 6

7.1: Empirical vs. Theoretical Two kinds of probability: Empirical: based on observation of experiments Theoretical: what should happen

7.1: Finding Probability number of times event has occurred total number of experiments done

7.1: Law of Large Numbers Law of Large Numbers: Probability applies to a large number of trials, not a single experiment. Ex: Baby gender. The probability of having a boy is 50%. My sister-in-law just had a girl (and is expecting another!). The probability works when applied to the whole population (large number of trials), not when applied to my sister-in-law (a single experiment).

7.1: Practice Problem p275 #17Animal Number Treated Dog45 Cat40 Bird15 Rabbit5 TOTAL105 P(next animal is a dog) =

7.2: Theoretical Probability number of favorable outcomes total number of possible outcomes

7.2: Important Facts Probability of an event that can’t happen is 0 Probability of an even that must happen is 1 Every probability is a number from 0 to 1. The sum of all probabilities for an experiment is 1.

7.2: Example 3 (a) P(drawing a 5) = 4 52 Can you reduce the fraction?

7.2: Example 3(b) P (drawing NOT a 5) = = NOTICE! P (drawing NOT a 5) = 1 - P (drawing a 5)

7.2: Practice Problems p 282 #21, 23 P(drawing a black card) = P(drawing a red card or a black card) =

7.2: Practice Problems p282 #27 P (red) = P (green) = P (yellow) = P (blue) = =

7.3: Odds Odds against an event = P(success) P(failure) Odds in favor of an event = P(success) P(failure)

7.3: Practice Problems p 291 #53 Odds against selling out = P(sells out) P(does not sell out) Odds against selling out = = 0.11

7.4: Expected Value To find expected value: For each outcome, multiply the probability times the value of that outcome. Add the results together for all possible outcomes.

7.4: Fair Price Fair Price = Expected Value + Cost to Play

7.4: Practice Problem p301 #57 (a) P(1) = 9/16 = P(10) = 4/16 = 0.25 P(20) = 2/16 = P(100) = 1/16 =

7.4: Practice Problem p301 #57 (b) Expected Value = $1* $10* $20* $100* = $

7.4: Practice Problem p301 #57 (c) Fair Price = Expected Value + Cost 0 = $ C C = $ (it makes sense to round to two decimal places)

7.5: Tree Diagrams Counting Principal: If there are M possible outcomes for a first experiment and N possible outcomes for a second experiment, there are M*N total possible outcomes. Ex: I have three shirts and two pairs of pants. I can make 3*2 = 6 outfits. The list of possible outcomes (outfits) is the “sample space”

7.5: Practice Problem p311 #11 (a) 2*2 = 4 p311 #11 (b) H T H T H T Sample Space HH HT TH TT

7.5 Practice Problem p311 #11 (c) 1/2 p311 #11 (d) 2/4 = 1/2 p311 #11 (e) 1/2

7.6: Or and And Problems P(A or B) = P(A) + P(B) - P(A and B) P(A and B) = P(A) * P(B) Mutually Exclusive: P(A and B) = 0

7.6: Example 1 P(Even or >6) = P(Even) + P(>6) - P(Even and >6) = 5/10 + 4/10 - 2/10 = 7/10

7.6: Practice Problem p323 #97 (a) No. The probability of the second event is affected by the outcome of the first. (b) (c) P(A and B) = P(A)*P(B) = * 0.04 =

7.6: Practice Problem p323 #97 (d) P(A and NOT B) = P(A)*P(NOT B) = 0.001*0.96 = (e) P(NOT A and B) = P(NOT A)*P(B) = 0.999*0.001 = (f) P(NOT A and NOT B) = P(NOT A)*P(NOT B) = 0.999*0.999 =