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Understanding Probability: Coin Tosses, Sample Spaces, and Calculations

Explore the meaning of probability through coin tosses, sample spaces, and probability calculations. Learn how to determine the probability of events and solve contest problems.

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Understanding Probability: Coin Tosses, Sample Spaces, and Calculations

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  1. Introduction Lecture 25 Section 6.1 Wed, Mar 22, 2006

  2. What 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%.

  3. 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.

  4. The 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.

  5. The Sample Space • Example • Procedure: Toss a coin. • Observed characteristic: Which side landed up. • Sample space = {H, T}

  6. The Sample Space • Example • Procedure: Roll a die. • Observed characteristic: Which number landed up. • Sample space = {1, 2, 3, 4, 5, 6}

  7. 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.

  8. 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|.

  9. 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%.

  10. 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?

  11. 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?

  12. 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?

  13. 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?

  14. 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?

  15. 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.

  16. 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?

  17. 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.

  18. The Monty Hall Problem • Use a simulation to determine the correct answer. • MontyHall.exe.

  19. 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 squares?

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