1. Binomial Probabilities Copyright © 2003, N. Ahbel

2. A Binomial Experiment has the following features: 1.There are repeated situations, called trials 2.There are only two possible outcomes, called success (S) and failure (F), for each trial 3.The trials are independent 4.Each trial has the same probability of success 5.The experiment has a fixed number of trials

3. Suppose that in a binomial experiment

4. Suppose that in a binomial experiment with n trials

5. Suppose that in a binomial experiment with n trials the probability of success is p in each trial

6. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p

7. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is:

8. P(k suc) = n C k. p k. q n-k

9. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures

10. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures

11. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures

12. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures

13. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures

14. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures

15. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures

16. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures

17. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures Each trial must be either a success or a failure

18. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures Each trial must be either a success or a failure so successes + failures = trials

19. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures Each trial must be either a success or a failure so successes + failures = trials k + failures = n

20. Suppose that in a binomial experiment with n trials the probability of success is p in each trial and the probability of failure is q, where q = 1-p then P(exactly k successes) or P(k suc) for short is: P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures Each trial must be either a success or a failure so successes + failures = trials k + failures = n failures = n - k

21. An example: Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that exactly one of the four patients currently being monitored is able to keep the cancer in remission for at least one year after chemotherapy. Find the probability that at least two patients are able to sustain remission for at least one year.

22. A Binomial Experiment has the following features: 1.There are repeated situations, called trials Each patient is a trial 2.There are only two possible outcomes, called success (S) and failure (F), for each trial Remission (S) and cancer reappears (F) 3.The trials are independent One patient’s outcome does not affect another's 4.Each trial has the same probability of success All patients are at the same risk level 5.The experiment has a fixed number of trials There are four patients in this group

23. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer.

24. Find the probability that exactly one of the four patients currently being monitored is able to keep the cancer in remission for at least one year after chemotherapy.

25. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that exactly one of the four patients currently being monitored is able to keep the cancer in remission for at least one year after chemotherapy. P(k suc) = n C k. p k. q n-k `

26. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that exactly one of the four patients currently being monitored is able to keep the cancer in remission for at least one year after chemotherapy. P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures

27. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that exactly one of the four patients currently being monitored is able to keep the cancer in remission for at least one year after chemotherapy. P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures P(1 suc) = 4 C 1. 0.7 1. 0.3 3

28. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that exactly one of the four patients currently being monitored is able to keep the cancer in remission for at least one year after chemotherapy. P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures P(1 suc) = 4 C 1. 0.7 1. 0.3 3 P(1 suc)  0.0756

29. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that exactly one of the four patients currently being monitored is able to keep the cancer in remission for at least one year after chemotherapy. P(k suc) = n C k. p k. q n-k P(k suc) = # of trials C # of suc. P(suc) # of suc. P(failure) # of failures P(1 suc) = 4 C 1. 0.7 1. 0.3 3 P(1 suc)  0.0756 About 8%

30. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer.

31. Find the probability that at least two patients are able to sustain remission for at least one year. P(k suc) = n C k. p k. q n-k

32. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that at least two patients are able to sustain remission for at least one year. P(k suc) = n C k. p k. q n-k P(2 suc) = 4 C 2. 0.7 2. 0.3 2  0.2646

33. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that at least two patients are able to sustain remission for at least one year. P(k suc) = n C k. p k. q n-k P(2 suc) = 4 C 2. 0.7 2. 0.3 2  0.2646 P(3 suc) = 4 C 3. 0.7 3. 0.3 1  0.4116

34. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that at least two patients are able to sustain remission for at least one year. P(k suc) = n C k. p k. q n-k P(2 suc) = 4 C 2. 0.7 2. 0.3 2  0.2646 P(3 suc) = 4 C 3. 0.7 3. 0.3 1  0.4116 P(4 suc) = 4 C 4. 0.7 4. 0.3 0  0.2401

35. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that at least two patients are able to sustain remission for at least one year. P(k suc) = n C k. p k. q n-k P(2 suc) = 4 C 2. 0.7 2. 0.3 2  0.2646 P(3 suc) = 4 C 3. 0.7 3. 0.3 1  0.4116 P(4 suc) = 4 C 4. 0.7 4. 0.3 0  0.2401 P(at least 2 suc) = P(2 suc) + P(3 suc) + P(4 suc)

36. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that at least two patients are able to sustain remission for at least one year. P(k suc) = n C k. p k. q n-k P(2 suc) = 4 C 2. 0.7 2. 0.3 2  0.2646 P(3 suc) = 4 C 3. 0.7 3. 0.3 1  0.4116 P(4 suc) = 4 C 4. 0.7 4. 0.3 0  0.2401 P(at least 2 suc) = P(2 suc) + P(3 suc) + P(4 suc) P(at least 2 suc)  0.2646 + 0.4116 + 0.2401  0.9163

37. Suppose that the probability for a certain cancer to remain in remission for at least one year after chemotherapy is 0.7 for all patients with that cancer. Find the probability that at least two patients are able to sustain remission for at least one year. P(k suc) = n C k. p k. q n-k P(2 suc) = 4 C 2. 0.7 2. 0.3 2  0.2646 P(3 suc) = 4 C 3. 0.7 3. 0.3 1  0.4116 P(4 suc) = 4 C 4. 0.7 4. 0.3 0  0.2401 P(at least 2 suc) = P(2 suc) + P(3 suc) + P(4 suc) P(at least 2 suc)  0.2646 + 0.4116 + 0.2401  0.9163 About 92%

38. Binomial Probabilities Copyright © 2003, N. Ahbel