You have a really annoying stapler that seems to randomly jam. On any given attempt to staple, it seems to independently jam 10.9% of the time. a) Out of 260 papers stapled, what is the probability that your stapler will jam less than 23 times? Round your answer to 4 decimals.

0.1230 = 12.30% probability that your stapler will jam less than 23 times

Step-by-step explanation:

I am going to use the normal approximation to the binomial distribution to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

The standard deviation of the binomial distribution is:

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that , .

In this problem, we have that:

So

Out of 260 papers stapled, what is the probability that your stapler will jam less than 23 times?

This is P(X < 23).

Using continuity correction, this is P(X < 23-0.5) = P(X < 22.5), which is the pvalue of Z when X = 22.5. So

has a pvalue of 0.1230

0.1230 = 12.30% probability that your stapler will jam less than 23 times