A geologist splits rocks to look for fossils.on average 10% of the rocks selected from aparticular area do infact contain fossils. The geologist selects a random sample of 20 rocks from this area, find the probability that:[i] Exactly one of the rocks contains fossils.[ii] At least one of the rocks contains fossils.
If we define a success as a rock containing a fossil; the probability of a success is 1/10. This then becomes a binomial random variable with parameters n=20 and p=1/10
so the probability of k successes out of n trials is
px(k)= (n choose k) p^k (1-p)^n-k
or for this case, px(1)=(20 choose 1) 1/10^1 (9/10)^19 = .2701 (approximately)
for ii) to find the probability of at least one success, we just need to find the probability of no successes in the 20 trials (which is equal to (9/10)^20) and find the complement of that. i.e. 1-(9/10)^20= .878 (approximately)