a: what is the probability that fewer than 5 employees steal?

b: more than 5 employees steal?

c: exactly 5 employees steal?

d: more than 5 but fewer than 15 employees steal?

Hi Megan. There are at least two different ways to get reasonable answers to this problem. The number of trials is fixed because there are 50 employees. There is exactly a 0.2 probability that any given employee steals. There are only two choices for each employee, steals or doesn’t steal (thief or non-theif). Thus, the data will follow a binomial distribution.

Calculating a binomial for 50 (n=50) trials is cumbersome but can be done. This might be the best approach for calculating the probability that “exactly 5 employees steal”. Fortunately, there is another approach. Because there are 50 trials, and the probability is not too close to either 0 or 1 (p=0.2), we can use the normal approximation of the binomial distribution. The normal curve which approximates this binomial distribution has a mean of np and a standard deviation of np(1-p).

Let’s start by using the binomial distribution to calculate the probability that “c: exactly 5 employees steal.”

Probability(5 thieves) = Probablity(45 nonthieves) = [50!/(5!•45!)](0.2)^{5}(1-0.2)^{45}

=0.030

The general formula is

Pr[K thives] = [50!/(K!•(50-K)!)](0.2)^{K}(0.8)^{50-K}

Now to solve for the other parts of the question with an exact answer, you have to perform this calculation repeatedly. For example, to solve a: what is the probability that fewer than 5 employees steal,” you need to do this for 0 thieves, 1 thief, 2 thieves, 3 thieves, and 4 thieves. Then you add those together. This is doable with an Excel spreadsheet. Otherwise, you might need to approximate with a normal distribution.

The approximating normal distribution has a mean of np = 50(0.2) = 10; SD = 10(1-0.2) = 8

We need to calculate the Z scores whenever using a normal distribution and consult a table to find the probability from the standard normal distribution. Z = (Y – 10)/8, where Y is the number of thieves.

For “a: what is the probability that fewer than 5 employees steal,” Y <= 4. So calculate Z for Y = 4. Z = (4-10)/8 =-0.75. Consulting the standard normal curve for Z = -0.75 gives us 0.2266. (If you calculate by the exact method using the binomial formula for thieves = 0,1,2,3,4, you get 0.018, which likeley means our approximation is not actually valid for this problem.)

I leave it to you to solve the remaining parts of the problem. Keep in mind what the normal distribution looks like. For part (a), we are looking at the left tail of the bell-shaped curve. For the other parts, you will need to subtract two probabilities which you get from the table from each other. For example, in (b), you will need to subtract the probability that there are 0,1,2,3,4, or 5 thieves from the probability that there are 0,1,2....49,50 thieves. (I will give you a hint: the probability that there are 0,1,2...49,50 thieves is 1).

I hope this helps you get started. Good luck.