Suppose Nokia store places 20 of its cell phones on a clearance sale, unknown to anyone 5 of these cell phones are defective. A customer selects 3 cell phones at random for inspection. Let X be the number of defective cell phones in the sample. Find the probability distribution of X?
My attempt:
$$
\begin{matrix}
x & 0 & 1 & 2 & 3 \\
P(X=x) & 0.015625 & 0.140625 & 0.421875 & 0.421875 \\
\\
\end{matrix}
$$
I calculated the values using the binomial distribution: $$
P(X = x) = nCx \ p^{n-x} (1-p)^x$$
Where, $n$=5, $x$=0,1,2,3 and $p$=(.25) [from 5/20]
Is this the correct way to do this?
Best Answer
No it is not correct.
In particular with only a quarter of the phones defective, getting three defectives out of three in the sample should have a low probability.
A second point is that the customer is presumably looking at three different phones (sampling without replacement), so you not be using the binomial distribution.
As an example, the probability of three phones defective, sampling without replacement, is $${3 \choose 3} \times \frac{5}{20} \times \frac{4}{19} \times \frac{3}{18} \approx 0.00877$$ rather than your ${3 \choose 3}0.25^0 0.75^3 = 0.421875.$