Binomials GrowingKnowing.com © 2011 GrowingKnowing.com © 2011

Binomial probabilities Your choice is between success and failure You toss a coin and want it to come up tails Tails is success, heads is failure Although you have only 2 conditions: success or failure, it does not mean you are restricted to 2 events Example: Success is more than a million dollars before I’m 30 Clearly there are many amounts of money over 1 million that would qualify as success Success could be a negative event if that is what you want Success for a student is find an error in the professor’s calculations If I am looking for errors, then I defined “success” as any event in which I find an error. GrowingKnowing.com © 2011

Conditions for binomials The outcome must be success or failure Mutually exclusive The probability of the event must be the same in every trial The outcome of one trial does not affect another trial. In other words, trials are independent If we take a coin toss, and you want tails for success. Success is tails, failure is heads Probability on every coin toss is 50% chance of tails It does not matter if a previous coin toss was heads or tails, chance of tails is still 50% for the next toss. Independent. GrowingKnowing.com © 2011

Don’t forget zero Would you like to clean my car or clean my shoes? Don’t forget zero as an option There are 3 possible outcomes: clean car, shoes, or nothing. If I toss a coin 3 times, what is the sample space? A sample space lists all the possible outcomes You could get tails on every toss of 3 (TTT). You could get tails twice and heads once (TTH) You could get tails once, and heads twice (THH) Did I miss anything? Do not forget you may get tails zero in 3 tries. (HHH) So the sample space is 3T 2T, 1T, and always include 0T GrowingKnowing.com © 2011

Excel function =BINOMDIST(successes, trials, probability, cumulative) Number of successes you want to measure Number of trials (how many times you try) Probability of each trial Cumulative is 0 for false, or 1 for true If you are doing a less-than, more-than, or between question, cumulative = 1 or TRUE Otherwise cumulative = 0 GrowingKnowing.com © 2011

Formulas is nCxpx(1-p)(n-x) From the last example: n=4, p=.25, x = 3 x=0 4C0p0(1-p)(4-0) = 1 (.250 (1-.25)4 = 1(.75) 4 = .31640625 x=1 4C1p1(1-p)(4-1) = 4(.251 (1-.25)4-1 = 1(.75) 3 = .421875 x=2 4C2p2(1-p)(4-2) = 6(.252 (.75)4-2 =6(.0625(.5625) = .2109375 x=3 4C3p3(1-p)(4-3) = 4(.253 (.75)1 = .0625(.75) 3 = .046875 x=4 4C4p4(1-p)(4-4) = 1(.254 (.75)0 = .003906(1) = .003906 Probability of guessing 3 successfully (x=3) is .046875 GrowingKnowing.com © 2011

How to calculate Let’s use an example to demonstrate. You are taking a multiple choice quiz with 4 questions. If you guess every question, what’s probability you guess 3 questions correctly? There are 4 choices for each question and 1 choice out of 4 is correct. Probability (p) to guess a question correctly is ¼ = .25 n is 4 because we have 4 trials. (questions on the quiz) x is 3, you are asked the probability of guessing 3 successfully. GrowingKnowing.com © 2011

Last example: trials=4, p= Last example: trials=4, p=.25, what is the probability you guess 3 questions correctly? x is the number of questions guessed correctly x=0 =binomdist(0,4,.25,0) = .316406 0 successes, 4 trials, .25 probability per trial, cumulate = false x=1 =binomdist(1,4,.25,0) = .421875 x=2 =binomdist(2,4,.25,0) = .210938 x=3 =binomdist(3,4,.25,0) = .046875 x=4 =binomdist(4,4,.25,0) = .003906 Probability of guessing 3 successfully (x=3) is .046875 GrowingKnowing.com © 2011

Sample questions Exact number of successes Calculation from the example : x=0, p = .316 x=1, p = .422 x=2, p = .211 x=3, p = .047 x=4, p = .004 Let’s use the findings from the last example to examine popular binomial questions. Exact number of successes What’s probability of guessing 3 questions correctly? =binomdist(3,4,.25,0) = .047 What’s probability of guessing 2 questions correctly? =binomdist(2,4,.25,0) = .211 What’s probability of guessing 0 questions correctly? =binomdist(0,4,.25,0) = .316 So we have 32% chance we’d guess no questions correctly GrowingKnowing.com © 2011

Calculation from the example : x=0, p = .316 x=1, p = .422 x=2, p = .211 x=3, p = .047 x=4, p = .004 Less What’s probability of guessing 2 or less questions correctly? We can work out and add up for each x (0,1,2) x=0 + x=1 + x=2 or (.316 + .422 + .211) = .949 Excel adds x for you if you set cumulative = 1 =Binomdist(2,4,.25,1) = .949 x is 2 4 is number of trails .25 is probability for each trial Cumulative is 1 or True, so Excel adds up values for x=0, x=1, and x=2 GrowingKnowing.com © 2011

Less Notice what is included and what is excluded. What’s probability guessing less than 2 questions correctly? =binomdist(1,4,.25,1) = .738 What’s probability guessing 2 or less questions correctly? =binomdist(2,4,.25,1) = .949 Notice what is included and what is excluded. Guessing “2 or less” we include x = 2. Guessing “less than 2” we exclude x = 2. GrowingKnowing.com © 2011

More What’s probability of guessing more than 2 questions correctly? Excel only cumulates from 0 up If you want higher than some middle number, use the complement rule. Accumulate up to but NOT including the x you want, then subtract from 1 to get the complement =1-binomdist(2,4,.25,1) = .051 Notice what is included and what is excluded. Guessing “2 or more” we include x = 2. Guessing “ more than 2” we exclude x = 2. Calculation from the example : x=0, p = .316 x=1, p = .422 x=2, p = .211 x=3, p = .047 x=4, p = .004 GrowingKnowing.com © 2011

More =1-binomdist(1,4,.25,1) = .262 =1-binomdist(0,4,.25,1) = .684 What’s probability guessing 2 or more questions correctly? =1-binomdist(1,4,.25,1) = .262 What’s probability of guessing at least 1 question correctly? =1-binomdist(0,4,.25,1) = .684 Note: ‘at least’ is a more-than question some students confuse ‘at least’ with ‘less-than’ GrowingKnowing.com © 2011

Between What’s probability of guessing between 2 and 4 (inclusive) questions correctly? We are told to include x=4 We want x=2, 3, 4 so .211+ +.047 + .004 = .262 Excel: think of 2 less-than questions and subtract Less than 4 (inclusive) =binomdist(4,4,.25,1) = 1.0 Less than 2 (inclusive) =binomdist(1,4,.25,1) = 0.7383 Subtract for the answer 1 - .738 = .262 To do the whole problem in one line in Excel =binomdist(4,4,.25,1) - binomdist(1,4,.25,1) = .262 What’s probability of guessing between 1 and 4 question correctly? If we assume 4 is inclusive. =binomdist(4,4,.25,1) – binomdist(0,4,.25,1) = .684 Calculation from the example : x=0, p = .316 x=1, p = .422 x=2, p = .211 x=3, p = .047 x=4, p = .004 GrowingKnowing.com © 2011

You need to practice because there are many ways of asking binomials questions which may confuse you the at first. Examples At least 3, Not less than 3 Greater than 2 None No more than 2 See the textbook for examples of how to interpret these different ways of asking binomial questions. GrowingKnowing.com © 2011