Probability and Matrices

[Guest]

A warehouse stores a large quantity of network cables: 60% of these cables have diameter 8mm, 20% have diameter 4mm. A sample of 10 cables is picked at random (because of the large number of cables stored, you can assume that the distribution of cable diameters (60-20-20) remains constant as the sample is picked). Answer the following:

i) What is the probability that the sample contains 5 cables with diameter 4mm?
ii) What is the probability that the sample contains 2 cables or more with diameter 4mm?
iii) What is the probability that the sample contains cables of only one diameter?

[HallsofIvy]

There appears to be missing data! You say "60% of these cables have diameter 8mm, 20% have diameter 4mm". That only adds up to 80%. What are the other 20%? You also say "(60-20-20)". Is there a third size that is another 20%?

If that is correct then we can treat (i) and (ii) as binomial distributions with the probability of "four mm" equal to 20% and "not four mm" equal to 60+ 20= 80%.
The probability of exactly 5 of 10 being "four mm" is $\begin{pmatrix}10 \\ 5\end{pmatrix} (0.2)^5(0.8 )^5$.

The probability of "2 or more four mm cables" is 1 minus the probability of "no four mm cables or 1 four mm cable".

The probability of no four mm cables is $\begin{pmatrix}10 \\ 0 \end{pmatrix}(0.2)^0 (0.8 )^{10}$.

The probability of exactly one four mm cable is $\begin{pmatrix}10 \\ 1\end{pmatrix}(0.2)^1(0.8 )^9$.