Binomials Growing Knowing com 2011 1 Binomial probabilities
Binomials Growing. Knowing. com © 2011 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 �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) �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
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 5
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 just guess every question what the probability is to guess 3 questions correctly. There are 4 choices for each question and 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. Growing. Knowing. com © 2011 6
�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 7
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? � Calculations from the example : x=0, p =. 316 x=1, p =. 422 x=2, p =. 211 x=3, p =. 047 x=4, p =. 004 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 � Growing. Knowing. com © 2011 8
�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 Calculation from the example : x=0, p =. 316 x=1, p =. 422 x=2, p =. 211 x=3, p =. 047 x=4, p =. 004 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. Growing. Knowing. com © 2011 9
�More Calculation from �What’s probability of guessing more than 2 the example : x=0, p =. 316 questions correctly? x=1, p =. 422 � Add x=3+ x=4 so (. 047 +. 004) =. 051 x=2, p =. 211 �What’s probability of guessing 2 or more x=3, p =. 047 questions correctly? x=4, p =. 004 � 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. Growing. Knowing. com © 2011 10
�More �What’s probability of guessing at least 1 question correctly? Calculation from the example : x=0, p =. 316 x=1, p =. 422 � Add x=1 + x=2 + x=3 + x=4 x=2, p =. 211 � Note: ‘at least’ is a more-than question x=3, p =. 047 � some students confuse ‘at least’ with ‘less-than’ x=4, p =. 004 �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 Growing. Knowing. com © 2011 11
�Between �What’s probability of guessing between 2 and 4 (inclusive) questions correctly? � We are told to include x=4 � So we want x=2, 3 and 4 so. 211+ +. 047 +. 004 = �What’s probability of guessing between 1 and 4 question correctly? Calculation from the example : x=0, p =. 316 x=1, p =. 422 x=2, p =. 211 x=3, p =. 047 x=4, p =. 004 � Assuming x=4 is inclusive, we want x=1+2+3+4 � x between 1 and 4 =. 422+. 211+. 047+. 004 =. 684 Growing. Knowing. com © 2011 12
�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. � Growing. Knowing. com © 2011 13
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