Probability for Binomial Distributions

Question

Directions: Find each specified probability for the given scenario.

The Sugarbear Candy Factory makes two types of chocolate candy bars- milk chocolate and milk chocolate with almonds. In a typical day, 60% of the candy bars being made are milk chocolate, and the rest are milk chocolate with almonds. At the end of the day, a quality control expert randomly chooses 14 chocolate bars for inspection.

a.) What is the probability that half of them contains almonds?

b.) What is the probability that at least half of them contain almonds?

Answer 1

Part A is a binomial probability problem in which the number of trials is equal to 14. The probability for a "success" which we are classifying as containing almonds is 40% or 0.40.

The probability that half of the 14 bars contain almonds means that x=7. The probability can be determined using the formula:

nCx(p)^x(1-p)^(n-x)

n=number of trials

x=number of successes

p=probability of a success

C is the combination function

It can also be determined using the binompdf function on a TI-84 (or similar) calculator.

binompdf(14,0.40,7)

Part B is also a binomial probability problem. Because we are in search of a probability that at least half contain almonds, the work will vary. What we want to know is: P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)+P(X=12)+P(X=13)+P(X=14). As you may have suspected, this would be a lot of work to do out whether we used the binomial formula or the binompdf function on the calculator. However, the calculator provides another function that can help us called binomcdf. Binomcdf stands for binomial cumulative distribution function. In using this function, we can determine P(X is less than or equal to 6). In subtracting this result from 1, we would be left with P(X is greater than or equal to 7):

1-binomcdf(14,0.40,6)