Binomials Growing Knowing com 2020 1 Binomial probabilities

Binomials Growing. Knowing. com © 2020 1

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. � Growing. Knowing. com © 2011 2

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. Growing. Knowing. com © 2011 3

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 3 T 2 T, 1 T, and always include 0 T Growing. Knowing. com © 2011 4

Excel function �=BINOM. DIST(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 Growing. Knowing. com © 2020 5

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. Growing. Knowing. com © 2011 6

Formula P(x) = n. Cxpx(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. Cx is the combination formula �many calculators give n. Cx with the push of a button Growing. Knowing. com © 2011 7

�Formulas is n. Cxpx(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 (. 250 (1 -. 25)4 = 1(. 75) 4 �x=1 1(1 -p)(4 -1) = 4(. 251 (1 -. 25)4 -1 = 1(. 75) 3 C p 4 1 �x=2 4 C 2 p 2(1 -p)(4 -2) = =. 31640625 =. 421875 6(. 252 (. 75)4 -2 =6(. 0625(. 5625) =. 2109375 3(1 -p)(4 -3) = 4(. 253 (. 75)1 =. 0625(. 75) 3 =. 046875 C p 4 3 �x=4 4 C 4 p 4(1 -p)(4 -4) = 1(. 254 (. 75)0 =. 003906(1) =. 003906 �x=3 �Probability of guessing 3 successfully (x=3) is. 046875 Growing. Knowing. com © 2011 8

�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 =binom. dist(0, 4, . 25, 0) =. 316406 � 0 successes, 4 trials, . 25 probability per trial, cumulate = false �x=1 �x=2 �x=3 �x=4 =binom. dist(1, 4, . 25, 0) =. 421875 =binom. dist(2, 4, . 25, 0) =. 210938 =binom. dist(3, 4, . 25, 0) =. 046875 =binom. dist(4, 4, . 25, 0) =. 003906 �Probability of guessing 3 successfully (x=3) is. 046875 Growing. Knowing. com © 2020 9

Sample questions Calculation from the example : x=0, p =. 316 �Let’s use the findings from the last example to x=1, p =. 422 x=2, p =. 211 examine popular binomial questions. x=3, p =. 047 x=4, p =. 004 �Exact number of successes �What’s probability of guessing 3 questions correctly? � =binom. dist(3, 4, . 25, 0) =. 047 �What’s probability of guessing 2 questions correctly? � =binom. dist(2, 4, . 25, 0) =. 211 �What’s probability of guessing 0 questions correctly? � =binom. dist(0, 4, . 25, 0) =. 316 � So we have 32% chance we’d guess no questions correctly Growing. Knowing. com © 2020 10

Calculation from the example : �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 x=0, p =. 316 x=1, p =. 422 x=2, p =. 211 x=3, p =. 047 x=4, p =. 004 �=Binom. dist(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 Growing. Knowing. com © 2020 11

Less �What’s probability guessing less than 2 questions correctly? � =binom. dist(1, 4, . 25, 1) =. 738 �What’s probability guessing 2 or less questions correctly? � =binom. dist(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. Growing. Knowing. com © 2020 12

�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 -binom. dist(2, 4, . 25, 1) Calculation from the example : x=0, p =. 316 x=1, p =. 422 x=2, p =. 211 x=3, p =. 047 x=4, p =. 004 =. 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. Growing. Knowing. com © 2020 13

More �What’s probability guessing 2 or more questions correctly? �=1 -binom. dist(1, 4, . 25, 1) =. 262 �What’s probability of guessing at least 1 question correctly? �=1 -binom. dist(0, 4, . 25, 1) =. 684 �Note: ‘at least’ is a more-than question � some students confuse ‘at least’ with ‘less-than’ Growing. Knowing. com © 2020 14

�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) � =binom. dist(4, 4, . 25, 1) = 1. 0 � Less than 2 (inclusive) � =binom. dist(1, 4, . 25, 1) = 0. 7383 � Subtract for the answer 1 -. 738 =. 262 �To do the whole problem in one line in Excel � =binom. dist(4, 4, . 25, 1) – binom. dist(1, 4, . 25, 1) =. 262 Calculation from the example : x=0, p =. 316 x=1, p =. 422 x=2, p =. 211 x=3, p =. 047 x=4, p =. 004 �What’s probability of guessing between 1 and 4 question correctly? If we assume 4 is inclusive. � =binom. dist(4, 4, . 25, 1) – binom. dist(0, 4, . 25, 1) =. 684 Growing. Knowing. com © 2020 15

�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. Growing. Knowing. com © 2011 16