1. iAcademy The Binomial Probability Distribution and Related Topics Foundational Statistics Lecture 9 Binomial probability distribution and its properties This lecture and its associated materials have been produced by Dr. Wittaya Kanchanapusakit (PhD, Cambridge) and Dr. Phanida Saikhwan (PhD, Cambridge) of iAcademy for the purposes of lecturing on the above described subject and the material should be viewed in this context. The work does not constitute professional advice and no warranties are made regarding the information presented. The Author and iAcademy do not accept any liability for the consequences of any action taken as a result of the work or any recommendations made or inferred. Permission to use any of these materials must be first granted by iAcademy.

2. iAcademy Agenda Review of week 8 Week 9 Lecture Material – Binomial probabilities – Additional properties of the binomial distribution

3. iAcademy Review of Week 8 The college student senate is sponsoring a spring break Caribbean cruise raffle. The proceeds are to be donated to the Samaritan Centre for the Homeless. A local travel agency donated the cruise, valued at $2,000. The students sold 2852 raffle tickets at $5 per ticket. Kevin bought six tickets. What is the probability that Kevin will win the spring break cruise to the Caribbean? What is the probability that Kevin will not win the cruise?

4. iAcademy Review of Week 8(2) Kevin bought six tickets. What is the probability that Kevin will win the spring break cruise to the Caribbean? What is the probability that Kevin will not win the cruise? Let’s discuss! – What is a variable, x? What are possible values of x?

5. iAcademy Review of Week 8(3) Let’s discuss! – Kelvin bought six tickets. What are possible outcomes? – What is probability that each ticket will win/lose?

6. iAcademy outcomeProbability LLLLLL WLLLLL LWLLLL LLWLLL LLLWLL LLLLWL LLLLLW Review of Week 8(4) Let’s discuss – What is the probability that Kevin will win the Spring break Caribbean cruise? Probability of each outcome is as follows:

7. iAcademy Review of Week 8(5) Note: – We can also determine the number of outcomes using the combination rule – When calculating probability that Kevin will win we can multiply the no. of outcome that one ticket will win with probability that one ticket will win

8. iAcademy Review of Week 8(6) Let’s discuss – What are Kevin’s expected earnings? – How much did Kevin effectively contribute to the Samaritan Centre for Homeless?

9. iAcademy Any Questions?

10. iAcademy Binomial probabilities Binomial probabilities are from binomial experiment. Binomial experiment or Bernoulli experiment is when there are exactly two possible outcomes (for each trial) of interest. – e.g. tossing a coin where outcomes are either head or tail

11. iAcademy Features of a binomial experiment There are a fixed number of trials, n. The n trials are independent and repeated under identical conditions. Each trial has only two outcomes: success (S) and failure (F). For each individual trial, the probability of success (p) is the same. Since only two outcomes, – p+q = 1; q = probability of failure The central problem of a binomial experiment is to find the probability of r success out of n trials.

12. iAcademy Examples of binomial trials Watch this video. Play Video: – Bernoulli trials in everyday life

13. iAcademy Any Questions?

14. iAcademy Let’s try analysing binomial experiment Blood type of 18 people selected randomly from the population was tested. Given that 9% of the population has blood type B. What is the probability that three of these 18 people have blood type B?

15. iAcademy Let’s try analysing Is this binomial experiment? – Yes, success = has type B blood and failure = does not have type B blood What are probabilities of success and failures? – 9% of the population has blood type B  p = 0.09 – q = 1 – p = 0.91 How many trials? – n = 18 We wish to compute the probability of 3 success out of 18 trials. – r = 3

16. iAcademy Computing probabilities for a binomial experiment(1)

17. iAcademy Let’s try

18. iAcademy Any Questions?

19. iAcademy Computing probabilities for a binomial experiment(2) Using a binomial distribution table (Table 3 of statistical tables given). The table gives the probability of – r successes in n independent trials, – each with probability of success p.

20. iAcademy Computing probabilities for a binomial experiment(3) e.g. n = 6 and p = 0.50 find P(4) by looking at the entry – in the row headed by 4 and – the column headed by 0.50 P(4) = 0.234

21. iAcademy Let’s try 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? n = 6, r = 4, p (success = grow) = 0.70, q = 1-0.70 = 0.30 Use binomial distribution table given, look for the section with n = 6, column headed by p = 0.70 and the row headed by r = 4. P(4) = 0.324

22. iAcademy Let’s try more … What is the probability that at least four seeds will germinate?

23. iAcademy Any Questions?

24. iAcademy Common expressions and corresponding inequalities ExpressionInequality Four or more successes At least four successes No fewer than four successes Not less than four successes Four or fewer successes At most four successes No more than four successes The number of successes does not exceed four

25. iAcademy Common expressions and corresponding inequalities(2) ExpressionInequality More than four successes The number of successes exceeds four Fewer than four successes The number of successes is not as large as four

26. iAcademy Any Questions?

27. iAcademy A rarely performed and somewhat risky eye operation is known to be successful in restoring the eyesight of 30% of the patients who undergo the operation. A team of surgeons has developed a new technique for this operation that has been successful in four of six operations. Does it seem likely that the new technique is much better than the old? How do you tell whether the new technique is better than the old one? Discussion: Find P(r)

28. iAcademy Discussion: Find P(r) Find the probability of at least four successes in six trials for the old technique. What are values of n, p, q and r? Use the formula: P(r) = C n,r p r q n-r to find P(4) What is P(4) using the table?

29. iAcademy Discussion: Find P(r) P(4) is not the answer as we are looking for the probability of at least four success out of the six trials or P(r ≥ 4). What should we find then?

30. iAcademy Discussion: Find P(r) Is the new technique better than the old? This means one of the following two things may happening:

31. iAcademy Any Questions?

32. iAcademy Graphing a binomial distribution How to graph a binomial distribution – Place r values on the horizontal axis. – Place P(r) values on the vertical axis. – Construct a bar over each r value extending from r - 0.5 to r + 0.5. The height of the corresponding bar is P(r)

33. iAcademy Let’s try graphing! Jim enjoys playing basketball. He figures that he makes about 50% of the field goals he attempts during a game. Make a histogram showing the probability that Jim will make 0, 1, 2, 3, 4, 5, or 6 shots out of six attempted field goals. This is a binomial experiment with – n = 6 trials and – p = 0.5

34. iAcademy rP(r) 3 4 5 6 Basketball problem Find P(r) values for n = 6 and p = 0.5 Use the given table rP(r) 00.016 10.094 20.234 Your turn to complete the table! 0.312 0.234 0.094 0.016

35. iAcademy Basketball problem(2) Use the values of P(r) to make a histogram rP(r) 3 4 5 6 r 00.016 10.094 20.234 0.312 0.234 0.094 0.016 The height of bar over r = P(r) = area

36. iAcademy Basketball problem(3) The graph is symmetrical! When p = 0.5, the graph of the binomial distribution will be symmetrical no matter how many trials we have.

37. iAcademy Any Questions?

38. iAcademy Mean and standard deviation

39. iAcademy What is the expected number of goals Jim will make? What is the standard deviation of the binomial distribution of the number of successful field goals Jim makes? Basketball problem(4) The mean is not only the balance point of the distribution but also the expected value of r.

40. iAcademy What is expected from binomial trials? What is standard deviation? Watch this video. Play Video: – Expected value and standard deviation of a binomial distribution

41. iAcademy Any Questions?

42. iAcademy Unusual values

43. iAcademy Any Questions?

44. iAcademy Summary A binomial experiment consists of a fixed number n of independent trials repeated under identical conditions. – Two outcomes for each trial: success and failure. – The probability p of success on each trial is the same. The number of successes r in a binomial experiment is the random variable for the binomial probability distribution. – The probabilities can be computed using a formula or using the given table or calculator.

45. iAcademy Summary(2)

46. iAcademy Tutorial Bring the given table (binomial probabilities) to the tutorial. Also bring the table to Lecture 10, you will need it for revision.

47. iAcademy