1. Expectation

2. OBJECTIVE
Find Expected Value of a Distribution. Find the probability of a Binomial distribution.

3. RELEVANCE
Be able to find probabilities of discrete random variables.

4. Definition……
Expected value of a discrete random variable is equal to the mean of the random variable.
It plays a role in decision theory (games of chance).
Note: Although probabilities can never be negative, expected value CAN be negative.

5. Formula……
Note: It is the same as the formula for the mean.

6. A Fair Game
In gambling games, an expected value of 0 implies that a game is a fair game (an unlikely occurrence!)
In a profit and loss analysis, an expected value of 0 represents the break-even point.
A game is fair if the expectation = 0

7. Example……
1000 tickets are sold at $1.00 each for a color TV valued at $350. What is the expected value of the gain if a person buys one ticket?
Set these up as gains minus losses.

8. 1000 tickets are sold at $1. 00 each for a color TV valued at $350
1000 tickets are sold at $1.00 each for a color TV valued at $350. What is the expected value of the gain if a person buys one ticket?
Answer:
Note: -$0.65 does not mean you lose 65 cents since the person can only win a TV valued at $350.
It means the average of the losses is $0.65 for each of the 1000 ticket holders.

9. Example
A ski resort loses $70,000 per season when it does NOT snow heavily and makes $250,000 profit when it DOES snow heavily. The probability of having a good season is 40%. Find the expectation of the profit.

10. A ski resort loses $70,000 per season when it does NOT snow heavily and makes $250,000 profit when it DOES snow heavily. The probability of having a good season is 40%. Find the expectation of the profit.
Answer:

11. Example
1000 tickets are sold at $1 each for 4 prizes of $100, $50, $25, and $10. What is the expected value if a person buys 2 tickets?

12. 1000 tickets are sold at $1 each for 4 prizes of $100, $50, $25, and $10. What is the expected value if a person buys 2 tickets?
Answer:

13. Example
At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

14. At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Answer:
Because the expected value is negative, you can expect to lose an average of $1.35 for each ticket that you buy.

15. Binomial Distributions

16. Many types of probabilities have 2 outcomes.
a. A coin: heads or tails
b. Baby Born: Male or Female
c. T/F Test: True or False
Some situations can be modified or reduced to 2 outcomes.
a. A medical treatment: effective or ineffective
b. Multiple choice test: correct or incorrect

17. Binomial Experiment satisfies 4 requirements…..
1. Has 2 outcomes or reduces to 2.
2. Fixed # of Trials
3. Outcomes for each trial must be
independent
4. The probability of success must
remain the same for each trial
This leads to…

18. Binomial Distribution…..
Definition – The outcomes of a binomial experiment and their corresponding probabilities.

19. Notation for the Binomial……
“n” – number of trials
“p” – the numerical probability of success
“q” – the numerical probability of failure;
q = 1 - p
“x” – the # of successes in “n” trials
x will always be a whole number – no decimals!

20. There are many ways to find the probability of a binomial.
One way is to use the formula below:

21. Let’s solve the following example using several different methods…..
Example: A coin is tossed 3 times. Find the probability of getting exactly 2 heads.
Notations Needed:
n = 3
p = ½
q = 1 – ½ = ½
x = 2

22. 1st Way: Using a tree diagram, find the sample space for 3 coins:
A coin is tossed 3 times. Find the probability of getting exactly 2 heads.
1st Way: Using a tree diagram, find the sample space for 3 coins:
HHH THH
HHT THT
HTH TTH
HTT TTT
There are 3 out of 8 possibilities of getting exactly 2 heads.
Probability = 3/8 or
0.375

23. 2nd way : Using the binomial formula.
A coin is tossed 3 times. Find the probability of getting exactly 2 heads.
2nd way : Using the binomial formula.
Remember…..
n = 3
p = ½
q = ½
x = 2

24. P. 711 in book (or look on copy) Answer: 0.375
A coin is tossed 3 times. Find the probability of getting exactly 2 heads.
3rd Way: Chart in Book
P. 711 in book (or look on copy)
Answer:

25. 4th and BEST way! – Graphing Calculator
A coin is tossed 3 times. Find the probability of getting exactly 2 heads.
4th and BEST way! – Graphing Calculator
Calculator Keys:
2nd Vars
0 or A: binompdf (n,p,x)
You will enter
binompdf(3, ½, 2)
Enter
Answer:

26. Example Answer: n = 20 p = .05 x = 5 0.002
Public Opinion Reported that 5% of Americans are afraid of being alone in the house at night. If a random sample of 20 Americans is selected, find the probability that there are exactly 5 people who are afraid of being alone in the house at night.
Answer:
n = 20
p = .05
x = 5
Binompdf(20, .05, 5)=
0.002

27. Example
A burglar alarm system has 6 fail-safe components. The probability of each failing is Find the probability that exactly 3 will fail.
Answer:
n = 6
p = 0.05
x = 3
Binompdf(6, 0.05, 3)=
0.002

28. Example
A student takes a random guess at 5 multiple choice questions. Find the probability that the student gets exactly 3 correct. Each question has 4 possible choices.
Answer:
n = 5
p = ¼
x = 3
Binompdf(5,1/4,3)=
0.088

29. Binomial Distributions…Continued
At Least
And
At Most

30. Example
Public Opinion reported that 5% of Americans are afraid of the dark. If a random sample of 20 is selected, find the probability that
A. At most 3 are afraid of the dark.
B. At least 3 are afraid of the dark.

31. This is the same as finding the probabilities for x = 0, 1, 2, and 3
Public Opinion reported that 5% of Americans are afraid of the dark. If a random sample of 20 is selected, find the probability that…..at most 3 are afraid of the dark.
This is the same as finding the probabilities for x = 0, 1, 2, and 3
Add the probabilities together.
n = 20
p = .05
x = 0, 1, 2, 3

32. The Long Way……. Using the calculator:
P(at most 3) = bpdf(20, .05, 0) + bpdf(20, .05, 1) + bpdf(20, .05, 2) + bpdf(20, .05, 3)
P(at most 3) = =

33. Is there a shorter way?....... Using your graphing calculator:
Put x’s in L1
Set a formula for L2.
bpdf(20, .05, L1)
The sum of L2 is your answer.

34. Public Opinion reported that 5% of Americans are afraid of the dark
Public Opinion reported that 5% of Americans are afraid of the dark. If a random sample of 20 is selected, find the probability that…..at least 3 are afraid of the dark.
n = 20
p = .05
x = 3, 4, 5, ………, 20

35. The answer is

36. You Try…..
A burglar alarm system has 6 fail-safe components. The probability of each failing is Find these probabilities.
a. Fewer than 3 will fail
b. None will fail
c. More than 3 will fail

37. A burglar alarm system has 6 fail-safe components
A burglar alarm system has 6 fail-safe components. The probability of each failing is Find these probabilities. A. Fewer than 3 will fail
n = 6
p = .05
x = 0, 1, 2
The answer is 0.998

38. You DO NOT need the lists for this one because there is only one x.
A burglar alarm system has 6 fail-safe components. The probability of each failing is Find these probabilities. B. None will fail
n = 6
p = .05
x = 0
The answer is 0.735
You DO NOT need the lists for this one because there is only one x.

39. A burglar alarm system has 6 fail-safe components
A burglar alarm system has 6 fail-safe components. The probability of each failing is Find these probabilities. C. More than 3 will fail
n= 6
p = .05
x = 4, 5, 6
The answer is

40. Assignment……
Worksheet