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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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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

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.

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

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

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)

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.

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) = ?

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.

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?

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?

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?

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.

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

Mean: Examples

- Mean household size
- Gambling: Roll a die
- A 6 occurs: you win $2.00
- Otherwise: you lose $1.00

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.

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

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

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

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

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 ???

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

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

Example

- Heights of young women
- Mean=?
- Standard Deviation=?
- First Quartile = ?
- Third Quartile = ?

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?

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.

The 68-95-99.7 Rule for Normal Distributions

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.

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

Homework

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

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