Binomial Distribution

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Binomial Distribution
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Recall that for a binomial distribution, we must
have

• Two possible outcomes, called success and failure

• Constant probability

• Independent trials

• A fixed number of trials

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Ex 1 Consider the random event with two
possible outcomes, such as heads and tails, from
tossing a coin.
Each toss is independent of every other toss and
probability is constant.
Suppose we define X to be the number of heads in
a series of 4 tosses.
What possible values can X take on?
X can be 0,1,2,3,4.
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We can find the probabilities by listing the
possible outcomes and counting
HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT THHH THHT
THTH THTT TTHH TTHT TTTH TTTT
X
0
1
2
3
4
P(X)
1/16
4/16
6/16
4/16
1/16
The binomial formula is
We can confirm the probabilities found above.
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We can find the probabilities by listing the
possible outcomes and counting
HHHH HHHT HHTH HHTT
HTHH HTHT HTTH HTTT
THHH THHT THTH THTT
TTHH TTHT TTTH TTTT


X 0 1 2 3 4
P(X) 1/16 4/16 6/16 4/16 1/16
The binomial formula is
We can confirm the probabilities found above.
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One thing that makes the binomial formula so nice
to use is that the graphing calculator has this
formula programmed already!
Lets try this on the calculator.
Enter 0,1,2,3,4 into L1. You can do this from
the homescreen or from within lists.
Enter the probabilities in L2 using the command
binompdf(4,0.5, L1).
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The values are the same as we found from listing
the outcomes or using the formula by hand.
We can use this same process to find a single
value.
To find the binomial probability for n 6, p
0.2, X 3, use the command binompdf(6,.2,3).
The probability is .08192.
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You should know how to use the formula, but as a
practical matter, the calculator is fine.
See that we can make a single calculation readily
using binompdf. Pdf stands for probability
density function. When dealing with discrete
distributions like the binomial, a calculation of
a pdf will give us a probability.
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When working with a continuous distribution, such
as the normal distribution, we used a cdf or
continuous density function, to find an area
under the curve.
We can also use a cdf with the binomial
distribution, but the calculator defines it a
little differently. It sums all the
probabilities for X 0 up to X the given value
of X.
If you want to find the probability for having up
to 4 girls in a family of 5 children, we have n
5, p .5, X 4.
Try the command binomcdf(5,.5,4). P(X 4)
.9687.
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Going back to the earlier example using the coin
toss, check your calculator. Have 0,1,2,3,4 in
L1 and the binompdf values in L2.
Now enter binomcdf(4,.5,L1) into L3.
Now lets look at the lists. The values of X are
in L1. The probability of each vales is in L2.
Now the sum of each probability (a cumulative
sum) is in L3.
In fact, the cumSum(L2) command will give the
same values as the binomcdf.
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When working a binomial problem I usually suggest
that students make a simple number line to aid in
finding which values to include. It sounds
unnecessary, but really it is often needed to
avoid mistakes.
Ex 2 Find the probability of getting at most 4
out of 10 true and false problems correct by
guessing (assume T and F are equally likely).
Answer The possible values of X are
0,1,2,3,,10.
At most 4 tells us that we could have
0,1,2,3,4.
Use the command binomcdf(10,.5,4).3769.
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Ex 3 Find the probability of making 5 or more
baskets out of 9 shots, for an 80 free throw
shooter in basketball.
Answer The possible values of X are
0,1,2,3,,9.
5 or more tells us that we could have
5,6,7,8,9.
To find this quantity we use the binomial command
on the calculator. 1- binomcdf(9,.8,4) .9804.
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Nex 4 Find the probability of missing 5 or more
baskets out of 9 shots, if you are an 80 free
throw shooter in basketball.
Answer The possible values of X are
0,1,2,3,,9.
Missing 5 or more means hitting only 0,1,2,3,or 4.
To find this quantity we use binomcdf(9,.8,4)
.01954.
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You are now ready to work problems on your own.
Make sure to draw numberlines whenever there are
inequalities. Youd be surprised at how easy it
is to make a mistake without the numberline.