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Chapter 7- Probability

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  1. Chapter 7- Probability • A phenomenon is random if any individual outcome is unpredictable, but each outcome tends to occur in a fixed proportion of a very long sequence of repetitions. Examples: Toss a coin Roll a pair of dice Sex of a newborn baby Draw one card from a deck Winning numbers in a lottery

  2. Definitions: • The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. It represents the frequency of occurrence of the outcomes • The sample space S of a random phenomenon is the list of all possible outcomes. • An event is any collection of outcomes in the sample space, i.e. a subset of the sample space.

  3. Examples: Find the sample space • Toss a coin once • Toss a coin 2 times • Toss 3 coins • Roll 1 die • Roll 2 dice: sum of the up faces/ difference of the up faces

  4. Probability Model • A probability model is a mathematical description of a random phenomenon consisting of two parts: • A sample space S • A way of assigning probabilities to events. • Example: Rolling a die • Notation: P(E) is the probability of an event E

  5. Probability Rules 1. The probability always satisfies 0<P(E)<1 2. The sum of the probabilities of all of the events in a sample space is 1. P(S) = 1 3. For any event E, we have P(not E) = 1 - P(E) 4. If 2 events A and B have no outcomes in common, then P(A or B) = P(A) + P (B)

  6. Probability Models • A probability model for a finite sample space, must satisfy the following conditions: • Each outcome has a probability that is between 0 and 1 • Probabilities of all outcomes in sample space must add up to 1.

  7. Equally Likely Outcomes • If a random experiment has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. • The probability of any event E, P(E) is • Example: P(roll a sum of 6) = ?

  8. Counting Principle • If there are m ways of doing one thing, n ways of doing a second thing, k ways of doing a third, …, then there are (m)(n)(k)…. ways of doing all three, one after the other.

  9. Examples: Counting • Example: How many code words of length 5 can be formed that use only these letters {a,b,c,d,e,f,g,h,i}? • A combination lock will open when the right choice of three numbers (from 1 to 40) is selected. How many different lock combinations are possible? • In Arizona, each automobile license plate number consists of 3-digit numbers followed by 3 letters. How many license plate numbers can be formed?

  10. More Examples: Probability • If the probability that Kerry gets an "A" in English class is 3/5, what is the probability that Kerry does not get an "A?" • Exactly one of three contestants will win a game show. The probability that Terry wins is 0.25 and the probability that Chris wins is 0.65. What is the probability that Toni wins? • Suppose three fair coins are tossed and the number of heads that appear is recorded. What is the probability of getting exactly two heads?

  11. More More Examples A pizza can be made with any of the following toppings: Cheese, pepperoni, mushrooms, ham, or olives. • How many different three-topping pizzas can be made? • What is the probability that a randomly created three-topping pizza will contain NO mushrooms (It’s very important to know the answer because I don’t like mushroom) • What is the probability that a randomly created three-topping pizza will contain bacon? Pepperoni?

  12. More More More Examples • Two dice are tossed. What is the probability that the total of the two dice is 7? • We wish to make a spinner that will be numbered 1 to 4, but will have the probability of spinning a "1" be 0.5. Draw the face for such a spinner.

  13. The Mean of a Probability Model • Grade Average • Suppose that the possible outcomes in a sample space S are numbers, and that is the probability of the outcome . Then the mean of this probability model is

  14. Mean: Examples • Mean household size • Gambling: Roll a die • A 6 occurs: you win $2.00 • Otherwise: you lose $1.00

  15. Probability Histogram

  16. Mean= Expected Value • The mean or expected value of an experiment basically tells you the overall average payoff if you play the game many times. • If you are gambling, the mean tells you about how much, on average, you are expected to win or lose each game.

  17. Law of Large Numbers • According to the Law of Large Numbers, as the random phenomenon is repeated a large number of times, • The proportion of trials on which each outcome occurs gets closer and closer to the probability of that outcome, and • The mean x of the observed values gets closer and closer to . Random in the sense of showing long-run regularity

  18. Roulette • A roulette wheel has 38 slots • 18 are black: odd • 18 are red: even • 2 are green:0 and 00 • A bet of $1 on red pay off an additional $1 if the ball lands in a red slot

  19. Distribution of average winnings • A gambler’s winnings in a night of 50 bets on red in roulette vary from night to night • The distribution of many nights’ results is approximately normal

  20. Sampling Distributions Revisited • Sampling variability is the fact that when we take repeated samples of the same size from the same population, results will vary from sample to sample. • We can represent the sampling distributions with histograms. • When examining sampling distributions, we want to look at the shape, center, and spread of the distributions

  21. A Random Sampling Experiment • Gallup asked a sample of 1523 people, “Please tell me whether or not you bought a state lottery ticket in the past 12 months” • Number of people answered “yes” is random; or the proportion of people in the sample said “yes” is random • If we drew 1000 different samples and repeated the questions, then we would find different proportions of people who said yes  By the Law of Large Number guarantees ???

  22. Sampling Distributions for sample size 100

  23. Sampling Distributions for sample size 1523

  24. Normal Curves that approximate the sampling Distributions

  25. Normal Curves • Normal curves are symmetric and bell-shaped with tails that fall off smoothly on either side, and have no outliers. The center of the normal curve is the center in several senses: Mean, median, center of symmetry. • A normal curve assigns probabilities to outcomes as follows: • the probability of an interval of outcomes is the area under the normal curve above that interval • The total area under any normal curve is exactly 1

  26. Normal Distributions • The mean of a normal distribution lies at the center of symmetry of the normal curve. • The standard deviation of a normal distribution is the horizontal distance from the mean to the point on the curve where the curve goes from being curved down to being curved up. • The first quartile of a normal distribution lies .67 of a standard deviation below the mean and the third quartile lies .67 of a standard deviation above the mean

  27. Example • Heights of young women • Mean=? • Standard Deviation=? • First Quartile = ? • Third Quartile = ?

  28. Example • This example also shows that why we only prefer to use mean and standard deviation instead of five-number summary • Questions: • How many percents of young women are shorter than 62.8 inches? • How many percents are between 62.8 and 68.2?

  29. 68-95-99.7 Rule • In a normal distribution: • 68% of the data lies within 1 standard deviation from the mean. • 95% of the data lies within 2 standard deviations from the mean. • 99.7% of the data lies within 3 standard deviations from the mean.

  30. The 68-95-99.7 Rule for Normal Distributions

  31. The Central Limit Theorem • A sample mean or sample proportion from n trials on the same random phenomenon has a distribution that is approximately normal when n is large. • The mean of this normal distribution is the same as the mean for a single trial. • The standard deviation of this normal distribution is the standard deviation for a single trial divided by the square root of n.

  32. Standard Deviation • Suppose that the possible outcomes • Of a sample space S are numbers and that is the probability of outcome . The variance of the probability model is • The standard deviation is the square root of the variance

  33. Homework • #3, 5, 7, 11, 13, 17, 20 • #19-35 odd, 41, 49, 55