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2. 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 © 20112

3. Conditions for binomials The outcome must be success or failure 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 © 20113

4. 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) 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 © 20114

5. Formula P(x) = n C x p x (1-p) (n-x) n is the number of trials. How often you tried to find a success event x is the number of successes you want p is the probability of success in each trial Remember that n C x is the combination formula many calculators give nCx with the push of a button GrowingKnowing.com © 20115

6. How to calculate Let’s use an example to demonstrate. You are taking a multiple choice quiz with 4 questions. You want to know if you guess every question, what’s probability you guess 3 questions correctly. There are 4 choices for each question of which one 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 © 20116

7. Formulas is n C x p x (1-p) (n-x) From the last example: n=4, p=.25, x = 3 x=0 4 C 0 p 0 (1-p) (4-0) = 1 (.25 0 (1-.25) 4 = 1(.75) 4 =.31640625 x=1 4 C 1 p 1 (1-p) (4-1) = 4(.25 1 (1-.25) 4-1 = 1(.75) 3 =.421875 x=2 4 C 2 p 2 (1-p) (4-2) = 6(.25 2 (.75) 4-2 =6(.0625(.5625) =.2109375 x=3 4 C 3 p 3 (1-p) (4-3) = 4(.25 3 (.75) 1 =.0625(.75) 3 =.046875 x=4 4 C 4 p 4 (1-p) (4-4) = 1(.25 4 (.75) 0 =.003906(1) =.003906 Probability of guessing 3 successfully (x=3) is.046875 GrowingKnowing.com © 20117

8. Sample questions Let’s use the findings from the example to examine popular binomial questions. Exact number of successes What’s probability of guessing 3 questions correctly? x=3, p =.047 What’s probability of guessing 2 questions correctly? X=2, p =.211 What’s probability of guessing 0 questions correctly? X=0, p =.316 So we have 32% chance we’d guess no questions right GrowingKnowing.com © 20118 Calculations from the example : x=0, p =.316 x=1, p =.422 x=2, p =.211 x=3, p =.047 x=4, p =.004

9. Less What’s probability of guessing less than 2 questions correctly? Add x=0 + x=1 so (.316 +.422) =.738 What’s probability of guessing 2 or less questions correctly? x=0 + x=1 + x=2 (.316 +.422 +.211) =.949 What’s probability of guessing less than 1 question correctly? X=0, p =.316 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 © 20119 Calculation from the example : x=0, p =.316 x=1, p =.422 x=2, p =.211 x=3, p =.047 x=4, p =.004

10. More What’s probability of guessing more than 2 questions correctly? Add x=3+ x=4 so (.047 +.004) =.051 What’s probability of guessing 2 or more questions correctly? x=2 + x=3 + x=4 (.211 +.047 +.004) =.262 Notice what is included and what is excluded. Guessing “2 or more” we include x = 2. Guessing “ more than 2” we exclude x = 2. GrowingKnowing.com © 201110 Calculation from the example : x=0, p =.316 x=1, p =.422 x=2, p =.211 x=3, p =.047 x=4, p =.004

11. More What’s probability of guessing at least 1 question correctly? Add x=1 + x=2 + x=3 + x=4 Note: ‘at least’ is a more-than question some students confuse ‘at least’ with ‘less-than’ You can always save time with the complement rule Calculate x for the small group, and if you subtract that probability by 1, you will get the rest of the grouping for x. 1-.316 =.684 is probability x=1 to 4 Double-check.422+.211+.047+.004 =.684 GrowingKnowing.com © 201111 Calculation from the example : x=0, p =.316 x=1, p =.422 x=2, p =.211 x=3, p =.047 x=4, p =.004

12. Between What’s probability of guessing between 2 and 4 (inclusive) questions correctly? We are told to include x=4 W want x=2, 3 and 4 so.211+ +.047 +.004 =.262 What’s probability of guessing between 1 and 4 question correctly? Assuming x=4 is inclusive, we want x=1+2+3+4 x between 1 and 4 =.422+.211+.047+.004 =.684 GrowingKnowing.com © 201112 Calculation from the example : x=0, p =.316 x=1, p =.422 x=2, p =.211 x=3, p =.047 x=4, p =.004

13. You need to practice as many of the ways of asking for binomials can be confusing until you’ve done a few Examples At least 4, Not less than 4 Greater than 4 None No more than 2 Binomials can take a long time with high numbers of trials, so use the complement rule to avoid excessive work. If trials are 30, probability is.5 per trial, what is the probability of less than 29 successes? Calculate x=30 and 29, add them, then subtract 1 to get x=0 to 28 Or, in 3 hours calculate each value from x=0 to x=28 and add them. GrowingKnowing.com © 201113