(a) A manufacturer produces light bulbs which are tested in the following way. A batch is accepted in either of the following cases: (i) a first sample of 5...

Answer Details

(a) With \(10\%\) faulty, \(P(\text{a bulb is good}) = 0.9\). For a sample of 5, \(P(\text{no faulty}) = (0.9)^5 = 0.59049\).

The batch is accepted if the first sample has no faulty, OR the first sample has at least one faulty but the second sample has none:

\[P(\text{accept}) = (0.9)^5 + \big[1 - (0.9)^5\big](0.9)^5 = 0.59049 + (0.40951)(0.59049).\]

\[= 0.59049 + 0.241813 = 0.832303 \approx 0.832.\]

(b) Bag of 15 marbles, 3 black, so \(P(\text{black}) = \tfrac{3}{15} = 0.2\), \(P(\text{not black}) = 0.8\), with replacement, \(n = 10\).

(i) Did not pick a black at all: \((0.8)^{10} = 0.1073742 \approx 0.107\).

(ii) Picked black at most three times \(= P(0)+P(1)+P(2)+P(3)\):

\[P(0)=0.107374,\ P(1)=\binom{10}{1}(0.2)(0.8)^9=0.268435,\]

\[P(2)=\binom{10}{2}(0.2)^2(0.8)^8=0.301990,\ P(3)=\binom{10}{3}(0.2)^3(0.8)^7=0.201327.\]

Sum \(= 0.879\) (to three decimal places).