IntroductionWhat is Probability?The Meaning of ProbabilityThe Sample SpaceSlide 5Slide 6Calculation of ProbabilityThe Probability of an EventExample: Probability of an EventExampleSlide 11Slide 12Slide 13Slide 14The Monty Hall ProblemSlide 16Slide 17Slide 18A Contest ProblemIntroductionLecture 25Section 6.1Wed, Mar 22, 2006What is Probability?A coin has a 50% chance of landing heads.What does that mean?The coin will land heads 50% of the time?•This is demonstrably false.The coin will land heads approximately 50% of the time?•Then the probability is approximately 50%, not exactly 50%.The Meaning of ProbabilityIt means that the fraction of the time that the coin lands heads will get arbitrarily close to 50% as the number of coin tosses increases without bound.This involves the notion of a limit as n approaches infinity.21##lim#tosseshea dstossesThe Sample SpaceAn experiment is a procedure that leads to an outcome.If at least one step in the procedure is left to chance, then the outcome is unpredictable.We observe a characteristic of the outcome.The sample space is the set of all possible observations.The Sample SpaceExampleProcedure: Toss a coin.Observed characteristic: Which side landed up.Sample space = {H, T}The Sample SpaceExampleProcedure: Roll a die.Observed characteristic: Which number landed up.Sample space = {1, 2, 3, 4, 5, 6}Calculation of ProbabilityWe will consider only finite sample spaces.If the n members of the sample space are equally likely, then the probability of each member is 1/n.ExamplesToss a coin, P(H) = 1/2.Roll a die, P(3) = 1/6.The Probability of an EventAn event is a collection of possible observations, i.e., a subset of the sample space.The probability of an event is the sum of the probabilities of its individual members.If the members of the sample space are equally likely, then P(E) = |E|/|S|.Example: Probability of an EventIn a full binary search tree of 25 values, what is the probability that a search will require 5 comparisons?Assume that all 25 values are equally likely.10 of them occupy the bottom row.Therefore, p = 10/25 = 40%.ExampleA deck of cards is shuffled and the top card is drawn.What is the probability that it isThe ace of spades?An ace?A spade?A black card?ExampleA deck of cards is shuffled, the top card is discarded, and the next card is drawn.What is the probability that it isThe ace of spades?An ace?A spade?A black card?ExampleA deck of cards is shuffled, the top card is drawn, and it is noted that it is red. Then the next card is drawn.What is the probability that it isThe ace of spades?An ace?A spade?A black card?ExampleA deck of cards is shuffled, the top card is drawn, and it is noted that it is black. Then the next card is drawn.What is the probability that it isThe ace of spades?An ace?A spade?A black card?ExampleTwo red cards and two black cards are laid face down.Two of them are chosen at random and turned over.What is the probability that they are the same color?The Monty Hall ProblemSee p. 301.There are three doors on the set for a game show. Call them A, B, and C. You get to open one door and you win the prize behind the door.One of the doors has a Ferrari behind it.You pick door A.The Monty Hall ProblemHowever, before you open it, Monty Hall opens door B and shows you that there is a goat behind it.He asks you whether you want to change your choice to door C.Should you change your choice or should you stay with door A?The Monty Hall ProblemThere are three plausible strategies.Stay with door A. •Door C still has a 1/3 chance, so door A must have a 2/3 chance.Switch to door C.•Door A still has a 1/3 chance, so door C must have a 2/3 chance.It doesn’t matter.•Both doors now have a 1/2 chance.The Monty Hall ProblemUse a simulation to determine the correct answer.MontyHall.exe.A Contest ProblemIf we choose an integer at random from 1 to 1000, what is the probability that it can be expressed as the difference of two