Question:
A piece of rope is made up of 100 strands. Assume that the breaking strength of the rope is the sum of the breaking strengths of the individual strands. Assume further that this sum may be considered to be the sum of an independent trials process with 100 experiments each having expected value of 10 pounds and standard deviation 1. Find the approximate probability that the rope will support a weight
Answer:
We want to know the probability that all ropes will fail below a certain limit, meaning that the sum of the weights at failure for each rope is below the indicated value. This is summarized in the following standardized sum formula:
\[{S}_{n}^{*} = \frac{{S}_{n}-n\mu}{\sqrt{n{\sigma}^{2}}}\]
The first example is right at the expected value so there are no surprises for the result:
z = (1000 - 100 * 10) / sqrt(100 * 1^2)
round(pnorm(z, lower.tail = FALSE), 4)## [1] 0.5Question:
Answer:
For the second example, the probability of the rope breaking below the expected value of 1000 is being sought. Therefore, we should expect something at least lower than 50%. We end up finding that the actual probability is much lower than that:
z = (970 - 100 * 10) / sqrt(100 * 1^2)
round(pnorm(z, lower.tail = FALSE), 4)## [1] 0.9987