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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

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• 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

• 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.

• 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

• 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

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)

• 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.

• 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) = ?

• 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.

• 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?

• 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?

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?

• 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.

The Mean of a Probability Model

• 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

• Mean household size

• Gambling: Roll a die

• A 6 occurs: you win \$2.00

• Otherwise: you lose \$1.00

• 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.

• 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

• 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

• 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

• 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

• 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 ???

• 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

• 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

• Heights of young women

• Mean=?

• Standard Deviation=?

• First Quartile = ?

• Third Quartile = ?

• 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?

• 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.

The 68-95-99.7 Rule for Normal Distributions

• 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.

• 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

• #3, 5, 7, 11, 13, 17, 20

• #19-35 odd, 41, 49, 55