Binomial probabilitiesYour 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 eventsExample: 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.
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Conditions for binomialsThe outcome must be success or failureThe probability of the event must be the same in every
trialThe 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 tailsIt does not matter if a previous coin toss was heads or
tails, chance of tails is still 50% for the next toss. Independent.
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Don’t forget zeroWould you like to clean my car or clean my shoes?
Don’t forget zero as an optionThere 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
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Excel function=BINOMDIST(successes, trials, probability,
cumulative)Number of successes you want to measureNumber 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 TRUEOtherwise cumulative = 0
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How to calculateLet’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
= .25n 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 6
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) = .3164060 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) = .046875x=4 =binomdist(4,4,.25,0) = .003906
Probability of guessing 3 successfully (x=3) is .046875
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Sample questionsLet’s use the findings from the last example to examine popular binomial questions.
Exact number of successesWhat’s probability of guessing 3 questions correctly?
=binomdist(3,4,.25,0) = .047What’s probability of guessing 2 questions correctly?
=binomdist(2,4,.25,0) = .211What’s probability of guessing 0 questions correctly?
=binomdist(0,4,.25,0) = .316 So we have 32% chance we’d guess no questions correctly
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Calculation from the example :
x=0, p = .316 x=1, p = .422 x=2, p = .211x=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
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Calculation from the example :
x=0, p = .316 x=1, p = .422 x=2, p = .211x=3, p = .047 x=4, p = .004
LessWhat’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.
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MoreWhat’s probability of guessing more
than 2 questions correctly?Excel only cumulates from 0 upIf 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.GrowingKnowing.com © 2011 11
Calculation from the example :
x=0, p = .316 x=1, p = .422 x=2, p = .211x=3, p = .047 x=4, p = .004
MoreWhat’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 questionsome students confuse ‘at least’ with ‘less-
than’GrowingKnowing.com © 2011 12
BetweenWhat’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
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Calculation from the example :x=0, p = .316 x=1, p = .422 x=2, p = .211x=3, p = .047 x=4, p = .004
You need to practice because there are many ways of asking binomials questions which may confuse you the at first.
ExamplesAt least 3, Not less than 3Greater than 2NoneNo more than 2
See the textbook for examples of how to interpret these different ways of asking binomial questions.
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