Answer:
A) 0.00686
B) 0.01673
C) 0.99587
Step-by-step explanation:
For this question, we are interested in the first success and we will Thus make use of geometric probability distribution.
It goes by the formula;
P(x) = [(1 - p)^(x - 1)]p
We are told that One out of every 119 tax returns that a tax auditor examines requires an audit.
Thus; p = 1/119 = 0.0084
A) Probability that the first return requiring an audit is the 25th return the tax auditor examines is given by;
P(25) = [(1 - 0.0084)^(25 - 1)]0.0084
P(25) = 0.00686
B) Probability the first requiring an audit is the first or second return the tax auditor examines, is;
P(1 or 2) = P(1) + P(2)
P(1) = [(1 - 0.0084)^(1 - 1)]0.0084
P(1) = 0.0084
P(2) = [(1 - 0.0084)^(2 - 1)]0.0084
P(2) = 0.00833
Thus;
P(1 or 2) = 0.0084 + 0.00833
P(1 or 2) = 0.01673
C) Probability that none of the first five returns the tax auditor examines require an audit is given by;
P(X > 5) = 1 - P(X ≤ 5)
Now,
P(X ≤ 5) is expressed as;
P(X ≤ 5) = P(1) + P(2) + P(3) + P(4) + P(5)
We know P(1) & P(2) already. Let's find P(3), P(4) and P(5)
P(3) = [(1 - 0.0084)^(3 - 1)]0.0084
P(3) = 0.00826
P(4) = [(1 - 0.0084)^(4 - 1)]0.0084
P(4) = 0.00819
P(5) = [(1 - 0.0084)^(5 - 1)]0.0084
P(5) = 0.00812
P(X ≤ 5) = 0.0084 + 0.00833 + 0.00826 + 0.00819 + 0.00812
P(X ≤ 5) = 0.00413
P(X > 5) = 1 - 0.00413
P(X > 5) = 0.99587
