Finding Binomial Probabilities with a Formula

By Consumer Dummies

Here, you get to practice finding binomial probabilities by using a formula. The following problems have a binomial random variable with p = 0.55. Use the following formulas for the binomial distribution for the problems.

image0.jpg

where

image1.jpg

and

n! = (n – 1)(n – 2)(n – 3) . . . (3)(2)(1)

Sample questions

  1. What is the probability of exactly one success in eight trials? Round your answer to four decimal places.

    Answer: 0.0164

    The formula for calculating a probability for a binomial distribution is

    image2.jpg

    Here,

    image3.jpg

    and n! means n(n – 1)(n – 2) . . . (3)(2)(1). For example 5! = (5)(4)(3)(2)(1) = 120; 2! = (2)(1) = 2; 1! = 1; and by convention, 0! = 1.

    To find the probability of exactly one success in eight trials, you need P(X = 1), where n = 8 (remember that p = 0.55 here):

    image4.jpg

    Rounded to four decimal places, the answer is 0.0164.

  2. What is the probability of exactly two successes in eight trials? Round your answer to four decimal places.

    Answer: 0.0703

    The formula for calculating a probability for a binomial distribution is

    image5.jpg

    Here,

    image6.jpg

    and n! means n(n – 1)(n – 2) . . . (3)(2)(1). For example 5! = (5)(4)(3)(2)(1) = 120; 2! = (2)(1) = 2; 1! = 1; and by convention, 0! = 1.

    To find the probability of exactly two successes in eight trials, you want P(X = 2), where n = 8 (remember that p = 0.55 here):

    image7.jpg

    Rounded to four decimal places, the answer is 0.0703.

  3. What is the probability of getting at least one success in eight trials? Round your answer to four decimal places.

    Answer: 0.9983

    The formula for calculating a probability for a binomial distribution is

    image8.jpg

    Here,

    image9.jpg

    and n! means n(n – 1)(n – 2) . . . (3)(2)(1). For example 5! = (5)(4)(3)(2)(1) = 120; 2! = (2)(1) = 2; 1! = 1; and by convention, 0! = 1.

    In this case, X is the number of successes in n trials. You want

    image10.png

    because “at least one” means the same as “one or more.” The easiest way to answer this question is to take 1 minus P(X = 0), because that’s the opposite and easier to find.

    image11.jpg

    Rounded to four decimal places, this answer is 0.017. Now, plug the value of P(X = 0) in the formula to find P(X > 0):

    image12.jpg

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