A candy shop is having a lucky draw to generate publicity. On average, \(4\%\) of candy bars produced contain a lucky draw ticket each.
The shop owner orders \(r\) candy bars on a particular day. The number of candy bars that contain a lucky draw ticket each is the random variable \(D\).
State, in the context of the question, two assumptions needed to model \(D\) by a binomial distribution.[2]
You are now given that \(D\) can be modelled by distribution \(\mathrm{B}(r, 0.04)\).
Find the value of \(r\) if the variance is \(1.92\).
The shop owner orders \(k\) candy bars on another day.
The probability that there are more than \(3\) lucky draw tickets among the \(k\) candy bars is at least \(0.34\). Determine the minimum value of \(k\).[3]
The lucky draw box contains five numbered vouchers. Two of the vouchers are numbered \(0\), the three others are numbered \(1\), \(2\) and \(4\) respectively. A voucher is taken one at a time, at random and without replacement, until the second voucher labelled \(0\) is taken out. The random variable \(A\) is the sum of the numbers on the vouchers taken.
State the possible values that \(A\) can take and determine the probability distribution of \(A\).[4]
A customer plays this lucky draw once and receives \(\$A\).
Find the probability that the customer receives at least \(\$5\), given that he has taken out at least \(4\) vouchers.[2]