Binomial Probability Models By the end of this set of lessons you should be able to do the following 1 2 Identify Bernoulli Trials and Binomial Process Calculate related probabilities for a binomial process Background You should like this process Even if you don t I know you need to appreciate its usefulness in business and science Let s haul out that time tested probability chestnut to start the thinking What is the probability that a family with three children will have two boys We have all the tools to handle this problem We can list the sample space bbb bbg bgb bgg gbb gbg ggb ggg We can count the successful results purely a matter of opinion There are three outcomes that have two boys So the probability is 3 in 8 Now let s try to handle this What is the probability that a family with ten children will have two boys I would prefer not to list the sample space It has 210 1024 possible outcomes We need a different strategy Let s go back to the first simpler problem and develop a better way Better because it will handle larger sample spaces Observe that our problem has only two possible outcomes b or g at each step Also notice that the probabilities step by step do not change They are independent The p b 0 5 at each birth We can use that to our advantage What we want to do now is to construct the simplest outcome that meets our criteria two boys The outcome is bbg The probability for that outcome is calculated using independence to the right 1 1 1 p bbg p b p b p g 2 2 2 2 1 1 1 p bbg p b 2 p g 2 2 Now we notice that this is very restrictive We can place the girl first second or last So there are three ways to place the girl Then there must be three ways to place the two boys also We recall from counting that we can calculate how many ways using combinations as 3 C2 We combine the basic probability calculation 2 p 2b 1g 3 C2 p b 2 p g with the number of ways the factors can be combined to get our final result 1 3 1 1 2 1 2 2 Let s try this process on the big problem two boys in 10 children First we see that the p b 0 5 at each birth Second we notice that we still only have two possibilities b or g as before Finally we know that independence still applies Arizona State University Department of Mathematics and Statistics 1 of 4 Binomial Probability Models Our simplest successful outcome puts the two boys first b b g g g g g g g g Then we calculate its probability as before p bbgggggggg p b p b p g p g p g p g p g p g p g p g 2 8 1 1 p 2b 8 g p b 2 p g 8 2 2 I took a little observational shortcut to get to the bottom line I also know that I have restricted my thinking to only one successful outcome boys first The boys could be in any of the ten positions So just as before we have to think of all combinations that include exactly two boys There are C possibilities 10 2 2 8 The total probability is p 2b and 8 g 10 C2 p b 2 p g 8 10 C2 1 1 45 2 2 1024 Let s flip this problem What is the probability that this family with ten children would have exactly eight girl and two boy children We can rearrange factors to see that it is 8 2 1 1 45 p 8 g and 2b 10 C8 p b 2 p g 8 10 C8 2 2 1024 Examining the three calculations we have done a formula pops up very quickly p m successes out of n trials n Cm p success m p failure n m m n m This formula is presented in various textbooks as b n m p n Cm p 1 p where p is probability of success for each of m trials and we want exactly n successes Then since there are only two complementary possibilities the probability of failure is 1 p The situation described above is usually called a Bernoulli Trial The model is called a Binomial Probability model Bernoulli Trial Defined Any time a single process creates the same outcome space of mutually exclusive outcomes then a series of the processes trials is a Bernoulli experiment The formula previously stated does an excellent job of simplifying our calculations The boy girl problems are Bernoulli trials Regardless of the number of children under consideration we expect to see only boys or girls and at least for our simplification the probability of a boy is one half As you will see in a more realistic boy girl Bernoulli trial process that probability is actually a simplification to introduce the concept 2 of 4 Arizona State University Department of Mathematics and Statistics 2 of 5 Binomial Probability Models More Examples Example 1 A teacher gives a ten question true false test What is the probability that a student will score exactly 80 by purely guessing the correct response Let s stop to think about what we consider success here It is not true or false It is a correct or incorrect problem We don t know for a fact that all the answers are true Since we are guessing our probability of guessing correctly is 50 We either guess right or guess wrong two equally likely options The problem is a paraphrase of the conditions for the 8 g 2 b problem Now though g is good correct and 8 2 1 1 45 b is bad incorrect the calculation is p 8 g and 2b 10 C8 p b 2 p g 8 10 C8 2 2 1024 Let s change the question Example 2 Assume 70 is passing What is the probability that by purely guessing on a ten problem T F quiz a student will pass the test The student passes with 7 or 8 or 9 or 10 correct responses The passing situations are mutually exclusive That allows us to add the individual probabilities because there is no possibility of an overlap in the probabilities The calculation is tedious but it is simple p pass b 10 7 5 b 10 8 5 b 10 9 5 b 10 10 5 Example 3 A teacher gives a 5 question multiple choice test Each problem has exactly one correct choice from among four possibilities Assume 70 is passing What is the probability that a student purely by guessing can score a passing grade A grim reality to pass at the 70 level you must get either 4 or 5 correct The problem is in the binomial world For each problem the probability of guessing correctly is 1 in 4 The probability of guessing incorrectly is …