1. Probability Distributions
Definitions
Discrete vs. Continuous
Mean and Standard Deviation
TI 83/84 Calculator
Binomial Distribution

2. Definitions
Random Variable: A variable that has a single numerical value that is determined by the chance of an outcome of an experiment.
Probability Distribution: A table, graph, or formula that shows all the possible outcomes and their probabilities.

3. Probability Distribution: Raffle
500 tickets are sold for a raffle at $10 each. There will be one $1000 grand prize and two $200 other prizes given. Write down the probability distribution table. X
990
190
-10
P(X)
1/500
2/500
497/500

4. Discrete vs. Continuous
Discrete: A random variable is discrete if it has a finite number of outcomes or a countable number of outcomes.
Continuous: A variable is continuous if it is not discrete.

5. Two Requirements
Let x be a discrete random variable. Then

6. Expected Value
Expected Value: If the experiment is run many many times, then it is very likely that the average value of x will be very close to the expected value.

7. Expected Value of a Coin
A coin toss where 0 represents landing heads and 1 represents landing tails has expected value If I flip a coin many many times then the average outcome is likely to be 0.5 (half heads and half tails).

8. Standard Deviation Standard Deviation:
The standard deviation measures an average distance from the mean if the experiment is run many many times.

9. Insurance
The insurance bet in 21 involves placing a bet, say $10. If the dealer has a 10, Jack, Queen, or King, the dealer pays the player $20. If not the dealer takes the $10.
Suppose you have a King and a Queen and the dealer has an Ace showing. Should you buy insurance?
Now you have a 2 and a 7, your friend has a pair of 5’s and the dealer has an Ace showing. Should you buy insurance?

10. Insurance
The insurance bet in 21 involves placing a bet, say $10. If the dealer has a 10, Jack, Queen, or King, the dealer pays the player $20. If not the dealer takes the $10. X 20
-10
P(X)
14/49
35/49

11. Insurance
The insurance bet in 21 involves placing a bet, say $10. If the dealer has a 10, Jack, Queen, or King, the dealer pays the player $20. If not the dealer takes the $10.
Interpretation: If you are in this insurance situation many many times and bet $10 each time, then it is very likely that your average earnings will be dollars per bet. This is a bad bet.

12. Insurance
The insurance bet in 21 involves placing a bet, say $10. If the dealer has a 10, Jack, Queen, or King, the dealer pays the player $20. If not the dealer takes the $10. X 20
-10
P(X)
16/47
31/47

13. Insurance
The insurance bet in 21 involves placing a bet, say $10. If the dealer has a 10, Jack, Queen, or King, the dealer pays the player $20. If not the dealer takes the $10.
Interpretation: If you are in this insurance situation many many times and bet $10 each time, then it is very likely that your average earnings will be twenty-one cents per bet. This is a wise bet.

14. Rolling Dice
Find and interpret the expected value and standard deviation for the random variable that represents the outcome of tossing a six-sided die.
1-Var Stats(L1,L2)

15. The Raffle
Find the expected value and standard deviation for the raffle example:
500 tickets are sold for a raffle at $10 each. There will be one $1000 grand prize and two $200 other prizes given. X
990
190
-10
P(X)
1/500
2/500
497/500

16. The Raffle X
P(X)
1/500
990
2/500
190
-10
497/500

17. Bidding on a Contract X P(X) 10,000 100,000 -700 0.05 0.01 0.94
A contractor has figured that bidding on a contract costs her $700. There is a 5% chance that she will win the contract and make a $10,000 profit on the project and there is a 1% chance that she will win and establish a long term working relationship with the client resulting in a total of $100,000 profit. Find and interpret the expected value and standard deviation. X
10,000
100,000
-700
P(X)
0.05
0.01
0.94

18. Bidding on a Contract
A contractor has figured that bidding on a contract costs her $700. There is a 5% chance that she will win the contract and make a $10,000 profit on the project and there is a 1% chance that she will win and establish a long term working relationship with the client resulting in a total of $100,000 profit. Find and interpret the expected value and standard deviation.
Conclusion: If the contractor bids on many, many projects then it is very likely that she will average a profit that is very close to $842 per bid. Moreover, the $10,235 standard deviation illustrates that she should be sure to have a lot of capitol in the bank since the actual profit for a bid is likely to be far from $842.

19. Definition of Binomial Distribution
Binomial Distribution: The distribution of the result of an experiment with
A fixed number of trials, n
The trials are independent
Each trial results in success or failure
The probability of success, p, is the same for each trial.

20. Binomial Distribution Example
Suppose 22% of the population is angry about the economy. If 20 randomly selected people are surveyed, what is the probability that the number who are angry about the economy is
exactly 5?
at most 4?
2nd VARS (DISTR):
A: P(=x)
binompdf(n,p,x)
B: P(<x)
binomcdf(n,p,x)

21. Binomial Distribution Example
0.22 angry
20 sampled
P(x = 5)?
P(x < 4)at most 4?
2nd VARS (DISTR):
A: P(=x)
binompdf(n,p,x)
B: P(<x)
binomcdf(n,p,x)

22. Binomial Distribution Example
Two percent of the world population has Down’s Syndrome. If 300 randomly selected people are surveyed, what is the probability that the number who have Down’s Syndrome is
exactly 6?
less than 5?
at least 7?
2nd VARS (DISTR):
A: P(=x)
binompdf(n,p,x)
B: P(<x)
binomcdf(n,p,x)

23. Binomial Distribution Example
0.02 Down Syn.
300 Surveyed
exactly 6?
less than 5?
at least 7?
2nd VARS (DISTR):
A: P(=x)
binompdf(n,p,x)
B: P(<x)
binomcdf(n,p,x)

24. Statistical Significance
47% of all high school students have had sex. Greenville High has a new sex ed. program. It reports that of the 50 students surveyed only 20 had sex.
Find the probability that 20 or fewer out of 50 students selected from all high schools have had sex. Is this statistically significant?
2nd VARS (DISTR):
A: P(=x)
binompdf(n,p,x)
B: P(<x)
binomcdf(n,p,x)

25. Statistical Significance
0.47 had sex
50 surveyed
P(x < 20)
2nd VARS (DISTR):
A: P(=x)
binompdf(n,p,x)
B: P(<x)
binomcdf(n,p,x)

26. Discrimination?
39% of California residents are Hispanic. A company with 25 workers employs only 5 Hispanics. Is this statistically significant? Hint: Find the probability of employing 5 or fewer out of 25.
2nd VARS (DISTR):
A: P(=x)
binompdf(n,p,x)
B: P(<x)
binomcdf(n,p,x)

27. Discrimination? 0.39 Hispanic 25 Workers P(x < 5) 2nd VARS (DISTR):
A: P(=x)
binompdf(n,p,x)
B: P(<x)
binomcdf(n,p,x)

28. Binomial Distribution Statistical Significance
Only 42% of CCC algebra students pass. Of the 65 LTCC algebra students, 38 of them passed. Is this statistically significant? Hint: Find the probability that at least 38 of 65 CCC students will pass.
2nd VARS (DISTR):
A: P(=x)
binompdf(n,p,x)
B: P(<x)
binomcdf(n,p,x)