Question

A gold digging company determines there is a 20 % chance of successfully mining gold in a given mining site.

i) What is the probability that the first successful extraction of gold comes from the third mining site?
ii) What is the probability that the third successful extraction of gold comes from the seventh mining site?
iii) What is the mean number of mining sites that must be mined to successfully extract gold 3 times?
iv) What is the variance of the same number of mining sites?

Answer 1

The probability that the first successful extraction of gold comes from the third mining site is 12.8%. The probability that the third successful extraction of gold comes from the seventh mining site is 2.6%. The mean number of mining sites that must be mined to successfully extract gold 3 times is 5.

Step-by-step explanation:

i) The probability that the first successful extraction of gold comes from the third mining site is 0.8 * 0.8 * 0.2 = 0.128 or 12.8%.

ii) The probability that the third successful extraction of gold comes from the seventh mining site is 0.8^6 * 0.2 = 0.026 or 2.6%.

iii) The mean number of mining sites that must be mined to successfully extract gold 3 times is 1/p, where p is the probability of success. So in this case, it is 1/0.2 = 5.

iv) The variance of the number of mining sites is given by (1-p)/p^2, where p is the probability of success. So in this case, it is (1-0.2)/0.2^2 = 20.