Binomial Probabilities Skill 24

Objectives List the defining features of a binomial experiment. Compute binomial probabilities using the formula P(r) = Cn,r prqn – r Use the binomial table to find P (r). Use the binomial probability distribution to solve real-world applications.

1. There are a fixed number of trials (n) When there are exactly two possible outcomes (for each trial) of interest. These problems are called binomial experiments, or Bernoulli experiments. 1. There are a fixed number of trials (n) 2. The trials are independent 3. 2 outcomes (success and failure) 4. For each trial the probability of success is the same 5. Find the probability of r successes out of n trials

Example-Binomial experiment On a TV game show, each contestant has a try at the wheel of fortune. The wheel has 36 slots one is gold. If the ball lands in the gold slot, the contestant wins $50,000. What is the probability that the game show will have to pay the fortune to three contestants out of 100? Is this a binomial experiment?

Example–Binomial experiment Solution: 1. Each of the 100 contestants has a trial at the wheel, so there are trials in this problem. 2. The trials are independent, since the result of one spin of the wheel has no effect on the results of other spins.

Example-Solution 3. We are interested in only two outcomes on each spin of the wheel: Lands on the gold, or it does not. Gold success (S) Not gold failure (F). In general, the assignment of the terms success and failure to outcomes.

Example–Solution 4. On each trial

Example–Solution 5. We want to know the probability of 3 successes out of 100 trials. It turns out that the probability the show will have to pay the fortune to r = 3 contestants out of 100 is about 0.23.

Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula Suppose you are taking a timed final exam. You have three multiple-choice questions left to do. Each question has four suggested answers, and only one of the answers is correct. You have only 5 seconds left to do these three questions, so you decide to mark answers on the answer sheet without even reading the questions.

This is a binomial experiment. Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula Find the probability that you get zero, one, two, or all three questions correct? This is a binomial experiment. Each question can be thought of as a trial, n = 3 trials The possible outcomes on each trial are success S, a correct answer failure F, a wrong answer

The Probabilities on each trial are P(S)=.25 P(F)=.75 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula There are eight possible combinations of S’s and F’s. SSS SSF SFS FSS SFF FSF FFS FFF To compute the probability of each outcome, use the Multiplication Law The Probabilities on each trial are P(S)=.25 P(F)=.75

Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula Compute the probability of each of the eight outcomes. Outcomes for a Binomial Experiment with n = 3 Trials

Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula Now we can compute the probability of r successes out of three trials for r = 0, 1, 2 or 3. Compute P(1)… *P(1) stands for the probability of one success. There are three outcomes that show one success. *The outcomes; SFF, FSF, and FFS.

Now, find P(0), P(2), and P(3). P(r) for n = 3 Trials, p = 0.25 Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula Now, find P(0), P(2), and P(3). P(r) for n = 3 Trials, p = 0.25

Formula for the binomial probability distribution 𝑷 𝒓 = 𝒏! 𝒓! 𝒏−𝒓 ! 𝒑 𝒓 𝒒 𝒏−𝒓 = 𝑪 𝒏,𝒓 𝒑 𝒓 𝒒 𝒏−𝒓

Solution: a. This is a binomial experiment with 10 trials. Example–Compute P(r) using the binomial distribution formula A survey showed that 59% of Internet users are somewhat concerned about the confidentiality of their e-mail. Based on this information, what is the probability that for a random sample of 10 Internet users, 6 are concerned about the privacy of their e-mail? Solution: a. This is a binomial experiment with 10 trials. Probability of success is 59%. Now find the probability of 6 successes.

Example–Solution Need to find n, p, q, and r There is a 25% chance that exactly 6 out of 10 internet users are concerned about the privacy of e-mail

Using a Binomial Distribution Table In many cases we will be interested in the probability of a range of successes. In such cases, we need to use the addition rule. Example; for n = 6 and p = 0.50, P(4 or fewer successes) = P (r  4) = P (r = 4 or 3 or 2 or 1 or 0) = P(4) + P(3) + P(2) + P(1) + P(0)

Using a Binomial Distribution Table Table 3 of Appendix II gives values of P (r) for selected p values and values of n through 20. To use the table, find the appropriate section for n, and then use the entries in the columns headed by the p values and the rows headed by the r values.

Using a Binomial Distribution Table Excerpt from Table 3 of Appendix II for n = 6

Using a Binomial Distribution Table Likewise, you can find other values of P (r) from the table. In fact, for n = 6 and p = 0.50, P (r  4) = P(4) + P(3) + P(2) + P(1) + P(0) = 0.234 + 0.312 + 0.234 + 0.094 + 0.016 = 0.890

Using a Binomial Distribution Table

What is the probability that exactly four seeds will germinate? Example–Using the binomial distribution table to find P (r) A biologist is studying a new hybrid tomato. It is known that the seeds of this hybrid tomato have probability 0.70 of germinating. The biologist plants six seeds. What is the probability that exactly four seeds will germinate? Solution: This is a binomial experiment with n = 6 trials. The probability for success on each trial is 0.70.

Example–Solution Excerpt from Table 3 of Appendix II for n = 6

Find the probability of four or more seeds germinating. Example–Using the binomial distribution table to find P(r) Find the probability that at least four seeds will germinate? Solution: Find the probability of four or more seeds germinating. Compute P (r  4). P (r  4) = P (r = 4 or r = 5 or r = 6) = P(4) + P(5) + P(6) Must find P(4), P(5) and P(6)

Example–Solution P(4) = 0.324 P(5) = 0.303 P(6) = 0.118 Now, compute P (r  4). P (r  4) = P(4) + P(5) + P(6) = 0.324 + 0.303 + 0.118 = 0.745

24: Binomial Probabilities Summarize Notes Homework Worksheet Quiz

Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula What is the probability of success on anyone question? There are four possible answers, Probability of a correct answer is 1 4 =.25 The probability q of a wrong answer is 𝑞=1−𝑝=1−.25=.75 𝑛 = 3, 𝑝 = 0.25, 𝑎𝑛𝑑 𝑞 = 0.75 Now, what are the possible outcomes in terms of success or failure for these three trials?

Computing Probabilities for a Binomial Experiment Using the Binomial Distribution Formula Since the outcomes are mutually exclusive, we add the probabilities. 𝑃(1)=𝑃(𝑆𝐹𝐹 𝑜𝑟 𝐹𝑆𝐹 𝑜𝑟 𝐹𝐹𝑆)=𝑃(𝑆𝐹𝐹)+𝑃(𝐹𝑆𝐹)+𝑃(𝐹𝐹𝑆) = pq2 + pq2 + pq2 = 3pq2 = 3(0.25)(0.75)2 = 0.422