Question

I was laying a stone patio at the cottage. The stones come in three different colors and are either sparkly or not. Each stone is a bit different in size so the time it takes to install a particular stone is random. Based on my research, installation should take about 2 minutes per stone, on average, with a standard deviation of about 30 seconds and is pretty close to a normal distribution.

In terms of color, about 55% of the stones are mostly pink, 25% being mostly grey, 15% being mostly black. The pink ones are typically matte with only 10% being sparkly; the grey and black ones are 60% and 75% sparkly respectively. The sparkly ones are my favorite because they really pick up the afternoon sun.

To maintain the natural pattern of stones, I always pick them at random.

1. 1.Should I be pleased with my accomplishments if I manage to install 25 stones in 70 minutes?
2. 2. If I get 20 stones installed, what is the probability that 8 or more of them will be matte?
3. 3. Assuming that 15 randomly selected stones were sparkly, what is the probability any of them are pink?

Answer 1

Before starting the solution, I would like to point out that data given in problem seems inconsistent, as the sum of percentage distribution of colors of the stone is not 100. It is 55+25+15 = 95%. So some adjustment needs to be done. I am presenting the solution with black = 20% instead of 15%. If any correction in values is required, please just change them and follow the steps as per below. That is, numers might change but process won't

There is no special accomplishment!! A 70 minute time frame is actually way beyond the normal expectation

This is the probability p, for a Binomial distribution of matt stones among n = 20 selecition. The required probability is that X ≥ 8

Computing this value definitely requires a calculator. As already mentioned, take into account the appropriate correction.