1. 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)
0.05 1
0.25 2
0.35 3
0.15 4 5

2. Find Mean & Standard Deviation:
x = # books read
P(x)
0.13 1
0.21 2
0.28 3
0.31 4
0.07

3. Ex. Find the mean Find the Standard Deviation
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

4. 500 raffle tickets are sold at $2 each. You bought 5 tickets
500 raffle tickets are sold at $2 each. You bought 5 tickets. What’s your expected winning if the prize is a $200 tv.?

5. There are four envelopes in a box
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?

6. A person selects a card from a deck. If it is a red card, he wins $1
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?

7. 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?

8. Linear Transformations
Section 6.2A

9. 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

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

11. Multiplying/Dividing by a constant multiplies the mean and the standard deviation.
𝜎 𝑥 aX =a∙ 𝜎 𝑥

12. Pete’s Jeep Tours offers a popular half-day trip in a tourist area
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

13. Pete’s Jeep Tours offers a popular half-day trip in a tourist area
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

14. 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 – So, what is the average profit that Pete makes? What is the standard deviation in profits?

15. 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
0.3 1
0.4 2
0.2 3
0.1

16. 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
0.3 1
0.4 2
0.2 3
0.1

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

18. Effects of Linear Transformation on the Mean and Standard Deviation if 𝒀=𝒂+𝒃𝑿.
𝜇 𝑦 =𝑎+𝑏 𝜇 𝑥
𝜎 𝑦 = 𝑏 𝜎 𝑥
*Shape remains the same.

19. Example: Three different roads feed into a freeway entrance
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.
Road
Mean # Cars 1
800 2
1000 3
600

20. Mean of the Sum of Random Variables
For any two random variables, X and Y, if 𝑇=𝑥+𝑦 then the expected value of T is
𝐸 𝑇 = 𝜇 𝑇 = 𝜇 𝑥 + 𝜇 𝑦

21. Ex: What is the standard deviation of the # of cars coming from each road onto the freeway.
Mean # Cars
St. Dev. 1
800
34.5 2
1000
42.8 3
600
19.3

22. Variance of the Sum of Random Variables
For any two random variables, X and Y, if 𝑇=𝑥+𝑦 then the variance of T is

24. Mean
st dev x 20 5 y 24 3

25. Mean
st dev x 20 5 y 24 3

26. Mean
st dev x 20 5 y 24 3

27. Mean
st dev x 20 5 y 24 3

28. Find: and 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

29. Homework
Worksheet