Question

When the number of trials, n, is large, binomial probability tables may not be available. Furthermore, if a computer is not available, hand calculations will be tedious. As an alternative, the Poisson distribution can be used to approximate the binomial distribution when n is large and p is small. Here the mean of the Poisson distribution is taken to be μ = np. That is, when n is large and p is small, we can use the Poisson formula with μ = np to calculate binomial probabilities; we will obtain results close to those we would obtain by using the binomial formula. A common rule is to use this approximation when n / p ≥ 500.

To illustrate this approximation, in the movie Coma, a young female intern at a Boston hospital was very upset when her friend, a young nurse, went into a coma during routine anesthesia at the hospital. Upon investigation, she found that 12 of the last 31,000 healthy patients at the hospital had gone into comas during routine anesthesias. When she confronted the hospital administrator with this fact and the fact that the national average was 5 out of 62,000 healthy patients going into comas during routine anesthesias, the administrator replied that 12 out of 31,000 was still quite small and thus not that unusual.

Note: It turned out that the hospital administrator was part of a conspiracy to sell body parts and was purposely putting healthy adults into comas during routine anesthesias. If the intern had taken a statistics course, she could have avoided a great deal of danger.)

(a) Use the Poisson distribution to approximate the probability that 12 or more of 31,000 healthy patients would slip into comas during routine anesthesias, if in fact the true average at the hospital was 5 in 62,000. Hint: μ = np = 31,000 (5/62,000) = 2.5. (Leave no cell blank. You must enter "0" for the answer to grade correctly. Do not round intermediate calculations. Round final answer to 5 decimal places.)

Answer 1

When the number of trials, n, is large, binomial probability tables may not be available.

Furthermore, if a computer is not available, hand calculations will be tedious.

As an alternative, the Poisson distribution can be used to approximate the binomial distribution when n is large and p is small.

Here the mean of the Poisson distribution is taken to be μ = np

That is, when n is large and p is small, we can use the Poisson formula with μ = np to calculate binomial probabilities , we will obtain results close to those we would obtain by using the binomial formula.

A common rule is to use this approximation when n / p ≥ 500.

To calculate : the probability that 12 or more of 31,000 healthy patients would slip into comas during routine anesthesias.

Now,

Let X be the random variable.

Hence X ~ Bin (n, p) ~ Bin (31000, p)

Here p = 5/62000.

By using Poisson distribution ,

Therefore the probability that 12 or more of 31,000 healthy patients would slip into comas during routine anesthesias is 0.00001