GrowingKnowing.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,

GrowingKnowing.com © 2011

1GrowingKnowing.com © 2011

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.


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


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


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


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


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



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.


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 =

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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


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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GrowingKnowing.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,