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Chapter 5

Chapter 5. The Binomial Probability Distribution and Related Topics. Table of Contents. 5.1 Introduction to Random Variables and Probability Distributions. 5.2 Binomial Probabilities. 5.3 Additional Properties of the Binomial Distribution.

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Chapter 5

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  1. Chapter 5 The Binomial Probability Distribution and Related Topics

  2. Table of Contents 5.1 Introduction to Random Variables and Probability Distributions 5.2 Binomial Probabilities 5.3 Additional Properties of the Binomial Distribution

  3. 5.1 Introduction to Random Variables and Probability Distributions Definition A quantitative variable x is a random variable if the value that x takes on in a given experiment or observation is a chance or random outcome.

  4. Definition • A discrete random variable can take on only a finite number of values or a countable number of values. • E.g., the number of items, a single die roll 5.1 Introduction to Random Variables and Probability Distributions

  5. Definition • A continuous random variable can take on any of the countless number of values in a line interval. • E.g., measurable items, such as length, duration, temperature, etc. 5.1 Introduction to Random Variables and Probability Distributions

  6. Definition A probability distribution is an assignment of probabilities to each distinct value of a discrete random variable or to each interval of values of a continuous random variable. 5.1 Introduction to Random Variables and Probability Distributions

  7. Features of the probability distribution of a discrete random variable 1. The probability distribution has a probability assigned to each distinct value of the random variable. 2. The sum of all the assigned probabilities must be 1. 5.1 Introduction to Random Variables and Probability Distributions

  8. 1,400 20,000 Example 1 n = Σf = 20,000 Rel. freq. = = 0.07 5.1 Introduction to Random Variables and Probability Distributions

  9. Example 1 0.30 0.20 Probability P(x) 0.10 0 1 2 3 4 5 6 Score x 5.1 Introduction to Random Variables and Probability Distributions

  10. P(5 or 6) = P(5) + P(6) = 0.08 + 0.02 = 0.10 Example 1 5.1 Introduction to Random Variables and Probability Distributions

  11. ──────── σ = √Σ(x – µ)²P(x) Definitions The mean and the standard deviation of a discrete probability distributionare found using these formulas: µ = ΣxP(x) For a discrete probability distribution, µ is also called the expected value. 5.1 Introduction to Random Variables and Probability Distributions

  12. Note We already have a function on the TI-83/84 to use for discrete probability distributions. We can use the STAT|CALC|1-Var Stats function like we would for grouped or weighted data by giving it the x values for the x-List and the P(x) values for the weight-List. 1-Var Statsx-List,P(x)-List 5.1 Introduction to Random Variables and Probability Distributions

  13. 5.2 Binomial Probabilities Features of a binomial experiment 1. There are a fixed number of trials. We denote this number by the letter n. 2. The n trials are independent and repeated under identical conditions. 3. Each trial has only two outcomes: success, denoted by S, and failure, denoted by F.

  14. Features of a binomial experiment 4. For each individual trial, the probability of success is the same. We denote the probability of success by p and that of failure by q. Since each trial results in either success or failure, p + q = 1 and q = 1 – p. 5. The central problem of a binomial experiment is to find the probability of r successes out of n trials. 5.2 Binomial Probabilities

  15. We can think of each trial with its corresponding success as being an object. Then we have n = 3 objects (S1, S2, S3). 5.2 Binomial Probabilities

  16. Then, in order to have r = 3 successes, there is exactly Cn,r = C3,3 = 1 combination, namely S1S2S3. 5.2 Binomial Probabilities

  17. In order to have r = 2 successes, there are exactly C3,2 = 3 combinations, namely S1S2, S1S3, and S2S3. 5.2 Binomial Probabilities

  18. In order to have r = 1 success, there are exactly C3,1 = 3 combinations, namely S1, S2, and S3. 5.2 Binomial Probabilities

  19. In order to have r = 0 successes, there is exactly C3,0 = 1 combination, namely the one with no selection of successes. 5.2 Binomial Probabilities

  20. n! P(r) = prqn – r = Cn,rprqn – r r!(n – r)! Formula for the binomial probability distribution 5.2 Binomial Probabilities

  21. Calculator function The TI-83/84 has a function to calculate the binomial probability distribution formula directly. DISTR|DISTR|binompdf(n,p,r) [2nd][VARS] This function calculates the probability of getting r successes out of n trials when the probability of success on each trial is p. 5.2 Binomial Probabilities

  22. Example 5 n = 10 p = 0.59 r = 6 [ENTER] binompdf(10,0.59,6) .2503034245 The probability that exactly 6 out of the 10 Internet users are concerned about the privacy of e-mail is about 25%. 5.2 Binomial Probabilities

  23. P([0,r]) Calculator function There will be many times when we will want to be able to quickly add up the probabilities for 0 successes up to r successes, i.e. P([0,r]). … 0 1 2 r – 1 r DISTR|DISTR|binomcdf(n,p,r) This function calculates the cumulative probability of getting from 0 successes to r successes out of n trials when the probability of success on each trial is p. 5.2 Binomial Probabilities

  24. … 0 1 2 r – 1 P([0,r – 1]) P([r,n]) P([0,n]) = 1 Calculator function There will also be many times when we will want to be able to quickly add up the probabilities for r successes up to n successes out of n trials, i.e. P([r,n]). … r r + 1 n Since [0,n] represents all the simple events in our sample space, we know that P([0,n]) = 1. 5.2 Binomial Probabilities

  25. … 0 1 2 r – 1 P([0,r – 1]) P([r,n]) P([0,n]) = 1 Calculator function There will also be many times when we will want to be able to quickly add up the probabilities for r successes up to n successes out of n trials, i.e. P([r,n]). … r r + 1 n Since P([0,r – 1]) and P([r,n]) together add up to P([0,n]) = 1, then P([r,n]) = 1 – P([0,r – 1]). 5.2 Binomial Probabilities

  26. … 0 1 2 r – 1 P([0,r – 1]) P([r,n]) P([0,n]) = 1 Calculator function There will also be many times when we will want to be able to quickly add up the probabilities for r successes up to n successes out of n trials, i.e. P([r,n]). … r r + 1 n With the TI-83Plus, for P([r,n]), we will use 1–binomcdf(n,p,r–1) 5.2 Binomial Probabilities

  27. Example 6 n = 6 p = 0.70 (a) What is the probability that exactly four seeds will germinate? (I.e., P(4)) binompdf(6,0.70,4) .324135 (b) What is the probability that at least four seeds will germinate? 1–binomcdf(6,0.70,4–1) .74431 5.2 Binomial Probabilities

  28. 5.2 Binomial Probabilities

  29. 5.3 Additional Properties of the Binomial Distribution How to graph a binomial distribution 1. Place rvalues on the horizontal axis. 2. Place P(r) values on the vertical axis. 3. Construct a bar over each rvalue extending from r − 0.5 to r + 0.5. The height of the corresponding bar is P(r).

  30. Example 7 binompdf(6,0.70,0) binompdf(6,0.70,1) binompdf(6,0.70,2) etc. 5.3 Additional Properties of the Binomial Distribution

  31. P(r) 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 1 2 3 4 5 6 r Example 7 5.3 Additional Properties of the Binomial Distribution

  32. ─── σ = √npq How to compute µ and σ for a binomial distribution µ = np µ is also called the expected number of successes for the random variable r 5.3 Additional Properties of the Binomial Distribution

  33. Note The TI-83/84 has the square root function. It can be accessed by pressing [2nd][x²]. Example 8 When entering a square root into the calculator, be sure to match the close parenthesis with the square root’s open parenthesis. E.g., √(6*0.7*0.3) 5.3 Additional Properties of the Binomial Distribution

  34. Rule of thumb Unusual values For a binomial distribution, it is unusual for the number of successes r to be higher than μ + 2.5σor lower than μ − 2.5σ. 5.3 Additional Properties of the Binomial Distribution

  35. Quota problems Quota problems are always stated in a basic logical form of: “Find the number of trials n that would need to be performed in order for the probability for there to be at least r successes is #.” Stated in its most basic mathematical form, the problems always ask to find the smallest number of trials n so that P([r,n]) ≥ #. 5.3 Additional Properties of the Binomial Distribution

  36. Quota problems Unfortunately, the only method we have to solve quota problems is by trial and error. Suppose we have been given the problem of finding the smallest number of trials n for which P(r ≥ m) ≥ #. Then we first pick an educated guess for n and plug it into the formula 1–binomcdf(n,p,m–1). If the result gives a probability that is too high, then we try a lower n. If the result is too low, then we try a higher n. We do this until we get the smallest n for which the probability exceeds #. 5.3 Additional Properties of the Binomial Distribution

  37. Note As a time-saving feature of the TI-83/84 calculator, you can press [2nd][ENTER] to display the previous entry for editing. This becomes a very convenient feature when you’re testing many n’s in quota problems. 5.3 Additional Properties of the Binomial Distribution

  38. Example 9 Since p = 0.65, and we want at least 4 successes, a good starting point for n could be 7 since µ = np = (7)(0.65) = 4.55. nP(r ≥ 4) 7 0.800 1–binomcdf(7,0.65,4–1) 10 0.974 1–binomcdf(10,0.65,4–1) 9 0.946 1–binomcdf(9,0.65,4–1) When n = 10, P(r ≥ 4) ≥ 0.95. But, when n = 9, P(r ≥ 4) < 0.95. So n = 10 trials is the smallest number of trials required so that the probability that the number of successes is at least 4 is itself at least 95%. 5.3 Additional Properties of the Binomial Distribution

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