Binomial Probabilities Copyright © 2003, N. Ahbel

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

Suppose that in a binomial experiment

Suppose that in a binomial experiment with n trials

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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.

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 `

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

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

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 P(1 suc) 

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 P(1 suc)  About 8%

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

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 

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  P(3 suc) = 4 C 

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  P(3 suc) = 4 C  P(4 suc) = 4 C 

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  P(3 suc) = 4 C  P(4 suc) = 4 C  P(at least 2 suc) = P(2 suc) + P(3 suc) + P(4 suc)

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  P(3 suc) = 4 C  P(4 suc) = 4 C  P(at least 2 suc) = P(2 suc) + P(3 suc) + P(4 suc) P(at least 2 suc)  

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  P(3 suc) = 4 C  P(4 suc) = 4 C  P(at least 2 suc) = P(2 suc) + P(3 suc) + P(4 suc) P(at least 2 suc)   About 92%

Binomial Probabilities Copyright © 2003, N. Ahbel