1. Binomial PDF and CDF Section 8.1.2

2. Starter 8.1.2 Five marbles are on a table. Two of them are going to be painted with a “W” and the rest will be painted with an “L”. How many ways are there to choose the two marbles to be painted “W”?

3. Objectives Use a formula to find the PDF of a binomial random variable Find the Cumulative Distribution Function (CDF) of a binomial random variable by adding entries in the PDF and by using the binomcdf command California Standard 7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.

4. The Probability of One Outcome Return to the dice game where my probability of winning is 1/3. If we play the game 5 times, what is the probability that I will win exactly twice? –Let’s use rules of probability, not binomcdf command If I win twice (probability 1/3) and lose 3 times (probability 2/3) I could express this as: win AND win AND lose AND lose AND lose –By the AND rule, we get (1/3)(1/3)(2/3)(2/3)(2/3) –In simpler terms: (1/3) 2 (2/3) 3 =.033 But if we review the full PDF we did yesterday, we see P(X=2) =.33, so what’s wrong?

5. How Many Ways to Win Twice? We just found P(WWLLL), but there are other ways to win twice: –LLLWW or LWLWL or … –How many ways are there? This is actually the same as the starter, where we chose two things out of 5. –Here we are choosing where to put the two W’s in any of 5 places So there are 5 C 2 (or 10) ways to win twice –This is an “OR’ case, so we add the 10 identical probabilities –Alternatively, we multiply by 10, so get 10 x.033 =.33

6. Generalizing to a Formula Remember that n is the number of trials and p is the probability of success –That means (1-p) is the probability of failure So the probability of x successes is p x The probability of (n-x) failures is (1-p) (n-x) The number of ways to get x successes is n C x So the probability of x successes in n trials must be: n C x (p) x (1-p) (n-x) Use the formula to find the probability of exactly 3 successes in 5 trials where p=1/3 5 C 3 (1/3) 3 (2/3) 2 =.16 –Notice that the two exponents add up to n

7. The Cumulative Distribution Function Suppose we change the bet in the dice game such that I win the money if I have AT MOST 3 successes (where success still means rolling a 1 or 2). –Now what is the probability I win? –I can win by having 0 successes in 5 trials OR 1 success OR 2 successes OR 3 successes. –So I should add those 4 probabilities to answer the question. This describes the CDF, where we state the cumulative probability of having at most X successes.

8. Writing the CDF from a PDF Write the PDF of the binomial random variable as we did before. X012345 P(X).13.33.16.04.004

9. Writing the CDF from a PDF Write the PDF of the binomial random variable as we did before. Add a third row to the table and fill with the cumulative totals from P(X) by adding. X012345 P(X).13.33.16.04.004 C(X)

10. Writing the CDF from a PDF Write the PDF of the binomial random variable as we did before. Add a third row to the table and fill with the cumulative totals from P(X) by adding. X012345 P(X).13.33.16.04.004 C(X).13

11. Writing the CDF from a PDF Write the PDF of the binomial random variable as we did before. Add a third row to the table and fill with the cumulative totals from P(X) by adding. X012345 P(X).13.33.16.04.004 C(X).13.46

12. Writing the CDF from a PDF Write the PDF of the binomial random variable as we did before. Add a third row to the table and fill with the cumulative totals from P(X) by adding. X012345 P(X).13.33.16.04.004 C(X).13.46.79.95.991

13. Now Answer The Question We asked the probability that I have at MOST 3 successes. Read the C(X) line under X = 3 to find the probability is.95 –This is the sum of the probabilities that I have 0 or 1 or 2 or 3 successes. Calculator solution: binomcdf(5, 1/3, 3) –Note that command gives P(at MOST X successes) Based on that result, what is the probability that I have at LEAST 4 successes? 1 –.95 =.05 New question: What is the probability I have at least 3 successes? 1 – CDF(X = 2) = 1 –.79 =.21

14. Find the CDF on Calculator Put the PDF of the dice problem in L 1 and L 2 as we did yesterday –seq(x, x, 0, 5)→L 1 –binompdf(5, 1/3)→L 2 Now put the CDF in L 3 using binomcdf –binomcdf(5, 1/3)→L 3 Set up STAT PLOT 1 to be a histogram based on L 1 and L 3 –Draw the graph and sketch it in your notes –Comment on how the shape differs from PDF –What is the height of the last bar?

15. Objectives Use a formula to find the PDF of a binomial random variable Find the Cumulative Distribution Function (CDF) of a binomial random variable by adding entries in the PDF and by using the binomcdf command California Standard 7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.

16. Homework Read page 421 – 427 Do problems 5(e), 8 – 10