Warm Up • How do I know this is a probability distribution?

• What is the probability that Mary hits exactly 3 red lights?

• What is the probability that she gets at least 4 red lights?

• What is the probability that she gets less than two?

• Find the mean & standard deviation.

x=# red lights p(x)

0 0.05

1 0.25

2 0.35

3 0.15

4 0.15

5 0.05

Find Mean & Standard Deviation:

x = # books read

P(x)

0 0.13

1 0.21

2 0.28

3 0.31

4 0.07

Ex.1. Find the mean2. Find the Standard Deviation3. Find the probability that x is within one

deviation from the mean.

x = possible winnings

P(x)

5 0.1

7 0.31

8 0.24

10 0.16

14 0.19

500 raffle tickets are sold at $2 each. You bought 5 tickets. What’s your expected winning if the prize is a $200 tv.?

There are four envelopes in a box. One envelope contains a $1 bill, one contains a $5, one contains a $10, and one a $50 bill. A person selects an envelope. Find the expected value of the draw. What should we charge for the game for it to be fair?

A person selects a card from a deck. If it is a red card, he wins $1. If it is a black card between or including 2 and 10, he wins $5. If it is a black face card, he wins $10, and if it is a black ace, he wins $100. Find the expectation of the game. What would it be if it cost $10 to play? What should I charge to make it a fair game?

On a roulette wheel, there are 38 slots numbered 1 through 36 plus 0 and 00. Half of the slots from 1 to 36 are red; the other half are black. Both the 0 and 00 slots are green. Suppose that a player places a simple $1 bet on red. If the ball lands in a red slot, the player gets the original dollar back, plus an additional dollar for winning the bet. If the ball lands in a different-colored slot, the player loses the dollar bet to the casino. What is the player’s average gain?

LINEAR TRANSFORMATIONSSection 6.2A

Remember – effects of Linear Transformations

• Adding or Subtracting a Constant• Adds “a” to measures of center and location• Does not change shape or measures of spread

• Multiplying or Dividing by a Constant• Multiplies or divides measures of center and location by “b”• Multiplies or divides measures of spread by |b|• Does not change shape of distribution

Adding/Subtracting a constant from data shifts the mean but doesn’t change the variance or standard deviation.

•

•

( )E X c E X c

( )Var X c Var X

Multiplying/Dividing by a constant multiplies the mean and the standard deviation.

( ) ( )E aX aE X

2( ) ( )Var aX a Var X

𝜎 𝑥 (aX )=a ∙𝜎 𝑥

Pete’s Jeep Tours offers a popular half-day trip in a tourist area. The vehicle will hold up to 6 passengers. The number of passengers X on a randomly selected day has the following probability distribution. He charges $150 per passenger. How much on average does Pete earn from the half-day trip?

# Passengers Prob

2 0.15

3 0.25

4 0.35

5 0.2

6 0.05

Pete’s Jeep Tours offers a popular half-day trip in a tourist area. The vehicle will hold up to 6 passengers. The number of passengers X on a randomly selected day has the following probability distribution. He charges $150 per passenger. What is the typical deviation in the amount that Pete makes?

# Passengers Prob

2 0.15

3 0.25

4 0.35

5 0.2

6 0.05

What if it costs Pete $100 to buy permits, gas, and a ferry pass for each half-day trip. The amount of profit V that Pete makes from the trip is the total amount of money C that he collects from the passengers minus $100. That is V = C – 100. So, what is the average profit that Pete makes? What is the standard deviation in profits?

A large auto dealership keeps track of sales made during each hour of the day. Let X = the number of cars sold during the first hour of business on a randomly selected Friday. Based on previous records, the probability distribution of X is shown below. Suppose the dealership’s manager receives a $500 bonus from the company for each car sold. What is the mean and standard deviation of the amount that the manager earns on average?

# cars sold Prob

0 0.3

1 0.4

2 0.2

3 0.1

Suppose the dealership’s manager receives a $500 bonus from the company for each car sold. To encourage customers to buy cars on Friday mornings, the manager spends $75 to provide coffee and doughnuts. Find the mean and standard deviation of the profit the manager makes.

# cars sold Prob

0 0.3

1 0.4

2 0.2

3 0.1

Variance of y = a + bx• Relates to slope.

22

2

2 2 2

2

2 2 2

var( ) var( )

y x

yb

x

yb

x

y b x

y b x

b

Effects of Linear Transformation on the Mean and Standard Deviation if .

=

*Shape remains the same.

Example: Three different roads feed into a freeway entrance. The number of cars coming from each road onto the freeway is a random variable with mean values as follows. What’s the mean number of cars entering the freeway.

RoadMean #

Cars

1 800

2 1000

3 600

Mean of the Sum of Random Variables

+

For any two random variables, X and Y, if then the expected value of T is

Ex: What is the standard deviation of the # of cars coming from each road onto the freeway.

RoadMean #

CarsSt. Dev.

1 800 34.5

2 1000 42.8

3 600 19.3

Variance of the Sum of Random Variables

For any two random variables, X and Y, if then the variance of T is

Mean st dev

x 20 5

y 24 3

x y

3 2x y

Mean st dev

x 20 5

y 24 3

x y

3x y

Mean st dev

x 20 5

y 24 3

x y

3 2x y

Mean st dev

x 20 5

y 24 3

x y

3x y

Find: and 3 2x y 3 2x y

x P(x) y P(y)

3 0.32 10 0.22

4 0.14 20 0.34

5 0.12 30 0.18

6 0.42 40 0.26

Homework

Worksheet