Let \(X\) be the proportion of \(40 \mathrm{~W}\) light bulbs that break when dropped onto a carpet from a height of \(1 \mathrm{~m}\). You have tried it out, and your probability distribution for \(X\) is now \(X \sim \beta_{(23,48)}\).

(a) What is the expected proportion of light bulbs \(\mu_{X}\) that break when dropped onto a carpet from a height of \(1 \mathrm{~m}\) ?

(b) What is the probability that the next light bulb you drop onto a carpet from a height of \(1 \mathrm{~m}\), will break?

(c) What is the probability that \(X\) is within \(5 \%\) of this value?