Concept of the binomial distribution, binomial PDF and CDF, expectation and variance of binomial distribution
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).
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.
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.
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:
Using a calculator, we find
P(X≤14)=binomcdf(20, 0.6, 14)=0.874401
P(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
If X∼B(n,p), then
and
Nice work completing Binomial Distribution, here's a quick recap of what we covered:
Exercises checked off
Concept of the binomial distribution, binomial PDF and CDF, expectation and variance of binomial distribution
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).
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.
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.
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:
Using a calculator, we find
P(X≤14)=binomcdf(20, 0.6, 14)=0.874401
P(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
If X∼B(n,p), then
and
Nice work completing Binomial Distribution, here's a quick recap of what we covered:
Exercises checked off