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Binomial Random Variable Problems on the Calculator

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How to find probabilities?

Let n = number of trials.

Let p = probability of success

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Calculator!

  • A Binomial PDF (Probability Density function) allows you to find the probability that X is any value in a binomial distribution. It is found in the Distribution Menu:
    • 2nd VARS 0: binompdf( .
    • Its form is: Binompdf(n, p, X). (There are 3 important variables: n is the number of observations, p is the probability of success, and X is the number of successes you want.
    • If you don’t specify X, it will give you the probability for all values of X, from 0 to n as a list.

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Examples

  • 1. We want to compare the probability of getting 3 heads from 5 tosses of a coin with 4 heads on 5 tosses.

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Examples

  • 1. We want to compare the probability of getting 3 heads from 5 tosses of a coin with 4 heads on 5 tosses.

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Examples

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  • 2. Bob takes a true-false test of 6 questions and has absolutely no idea of any of the answers so he guesses on all of them. If 4 questions correct is passing, what is the probability that he passes the exam?
    • this is a binomial distribution with n = 6 and p = .5 and we need to add the probabilities of getting 4, 5, or 6 questions correct.

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Examples

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  • 2. Bob takes a true-false test of 6 questions and has absolutely no idea of any of the answers so he guesses on all of them. If 4 questions correct is passing, what is the probability that he passes the exam?
    • this is a binomial distribution with n = 6 and p = .5 and we need to add the probabilities of getting 4, 5, or 6 questions correct.

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Cumulative Binomial Probability

  • A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).
  • Ex: In a particular city, 63% of the adults own their home and 37% rent. A sample of 20 adults is taken. Find the probability that the sample will have at least half home-owners.
    • This is binomial with n = 20 and p = .63. We want the value of X = 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.
    • That is a lot of work, even with the Binompdf function!

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Calculator

  • To solve it, we turn to the Binomcdf formula found in the same menu. This gives the cumulative probabilities starting at X = 0. For instance, Binomcdf(20,.63,3) would give P(X = 0) + P(X =1) + P(X = 2) + P(X = 3) .
    • In our case we can find the sum of the probabilities that X = 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 then subtract that from 1. That will give us the probability that X = 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, or 20.

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More Examples

  • 1. What is the probability of obtaining 45 or fewer heads in 100 tosses of a coin?
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More Examples

  • 1. What is the probability of obtaining 45 or fewer heads in 100 tosses of a coin?
    • The sum of all these probabilities is the answer we seek: b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + . . . + b(x = 45; 100, 0.5)
    • binomcdf(100,.5,45) = 0.184 *try on calc!
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More Examples

2. The probability that a student is accepted to a prestigious college is 0.3. If 5 students from the same school apply, what is the probability that at most 2 are accepted?

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More Examples

2. The probability that a student is accepted to a prestigious college is 0.3. If 5 students from the same school apply, what is the probability that at most 2 are accepted?

    • b(x < 2; 5, 0.3) = 0.8369 on calc: binomcdf(5, .3, 2)