Finding Binomial Probabilities with a Formula
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.
where
and
n! = (n – 1)(n – 2)(n – 3) . . . (3)(2)(1)
Sample questions

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
Here,
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):
Rounded to four decimal places, the answer is 0.0164.

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
Here,
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):
Rounded to four decimal places, the answer is 0.0703.

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
Here,
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
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.
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):
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