1. Binomials
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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.
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3. 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.
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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)
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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5. Excel function
=BINOMDIST(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
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6. Formulas is nCxpx(1-p)(n-x)
From the last example: n=4, p=.25, x = 3
x=0 4C0p0(1-p)(4-0) = 1 (.250 (1-.25)4 = 1(.75) =
x=1 4C1p1(1-p)(4-1) = 4(.251 (1-.25)4-1 = 1(.75) =
x=2 4C2p2(1-p)(4-2) = 6(.252 (.75)4-2 =6(.0625(.5625) =
x=3 4C3p3(1-p)(4-3) = 4(.253 (.75)1 = .0625(.75) 3 =
x=4 4C4p4(1-p)(4-4) = 1(.254 (.75)0 = (1) =
Probability of guessing 3 successfully (x=3) is
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7. 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.
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8. Last example: trials=4, p=
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) =
0 successes, 4 trials, .25 probability per trial, cumulate = false
x=1 =binomdist(1,4,.25,0) =
x=2 =binomdist(2,4,.25,0) =
x=3 =binomdist(3,4,.25,0) =
x=4 =binomdist(4,4,.25,0) =
Probability of guessing 3 successfully (x=3) is
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9. Sample questions Exact number of successes
Calculation from the example :
x=0, p = .316 x=1, p = .422 x=2, p = .211
x=3, p = .047 x=4, p = .004
Let’s use the findings from the last example to
examine popular binomial questions.
Exact number of successes
What’s probability of guessing 3 questions correctly?
=binomdist(3,4,.25,0) = .047
What’s probability of guessing 2 questions correctly?
=binomdist(2,4,.25,0) = .211
What’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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10. Calculation from the example :
x=0, p = .316 x=1, p = .422 x=2, p = .211
x=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 ( ) = .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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11. Less Notice what is included and what is excluded.
What’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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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-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.
Calculation from the example :
x=0, p = .316 x=1, p = .422 x=2, p = .211
x=3, p = .047 x=4, p = .004
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13. More =1-binomdist(1,4,.25,1) = .262 =1-binomdist(0,4,.25,1) = .684
What’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 question
some students confuse ‘at least’ with ‘less-than’
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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 = .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) =
Subtract for the answer = .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 = .211
x=3, p = .047 x=4, p = .004
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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.
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