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Binomial Distribution
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Concept of the binomial distribution, binomial PDF and CDF, expectation and variance of binomial distribution
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The Concept of a Binomial Distribution
SL 4.8
The binomial distribution models situations where the same action is repeated multiple times, each with the same chance of success. It has two key numbers: the number of attempts (n) and the probability of success in each attempt (p).
If a random variable X follows a binomial distribution, we write X∼B(n,p).
Binomial PDF with calculator
SL 4.8
The binomial probability density function (aka pdf) is a function that models the likelihood of obtaining k successes from n trials where the likelihood of success of each trial is p. We calculate the probability of exactly k successes in n trials, P(X=k), using the calculator's binompdf function.
Press 2nd → distr → binompdf(. Once in the binompdf function, write your n value after "trials," your p value after "p," and your k value after "x value." Then hit enter twice and the calculator will return the probability you are interested in.
The distr button is located above vars . Once in the distr menu, you can also click alpha → A to navigate to the binompdf function.
Binomial CDF with Calculator
SL 4.8
The binomial cumulative density function tells us the probability of obtaining k or fewer successes in n trials, each with a likelihood of success of p. We calculate the probability of less than or equal to k successes in n trials, P(X≤k), using the calculator's binomcdf function.
Press 2nd → distr → binomcdf(. Once in the binomcdf function, write your n value after "trials," your p value after "p," and your k value after "x value." Then hit enter twice and the calculator will return the probability you are interested in.
The distr button is located above vars . Once in the distr menu, you can also click alpha → B to navigate to the binomcdf function.
Example
A student is taking a 20 question multiple choice exam where each question is worth 1 point. The student needs to score 11 points for a 5, and 15 points for a 6.
Given that the probability the student answers each question correctly is 0.6, find the probability that he scored a 5.
Let X∼B(20,0.6) be the student's score. The student scores a 5 if 11≤X<15 (i.e. 11≤X≤14). We can express this probability as the difference of two probabilities:
P(11≤X≤14)=P(X≤14)−P(X≤10)
Using a calculator, we find
P(X≤14)=
binomcdf(20, 0.6, 14)=0.874401P(X≤10)=
binomcdf(20, 0.6, 10)=0.244663
Subtracting we find P(11≤X≤14)=0.630.
Note that you get the same result from doing binompdf(20, 0.6, 11) + binompdf(20, 0.6,12) + binompdf(20, 0.6, 13) + binompdf(20, 0.6, 14)
We could visualize this on a graph as
Expectation and Variance of Binomial Distribution
SL 4.8
If X∼B(n,p), then
E(X)=np📖
and
Var(X)=np(1−p)📖
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