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
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)
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. Formula P(x) = nCxpx(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 nCx is the combination formula
many calculators give nCx with the push of a button
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6. 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.
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7. 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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8. Sample questions x=0, p = .316 x=1, p = .422 x=2, p = .211
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?
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
Calculations 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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9. Notice what is included and what is excluded.
Less
What’s probability of guessing less than 2 questions correctly?
Add x=0 + x=1 so ( ) = .738
What’s probability of guessing 2 or less questions correctly?
x=0 + x=1 + x=2 ( ) = .949
What’s probability of guessing less than 1 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.
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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10. Notice what is included and what is excluded.
More
What’s probability of guessing more than 2 questions correctly?
Add x=3+ x=4 so ( ) = .051
What’s probability of guessing 2 or more questions correctly?
x=2 + x=3 + x=4 ( ) = .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.
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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11. More What’s probability of guessing at least 1 question correctly?
Add x=1 + x=2 + x=3 + x=4
Note: ‘at least’ is a more-than question
some students confuse ‘at least’ with ‘less-than’
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.
=.684 is probability x=1 to 4
Double-check = .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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12. 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 =
What’s probability of guessing between 1 and 4 question correctly?
Assuming x=4 is inclusive, we want x=
x between 1 and 4 = = .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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13. 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.
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