At a particular checkout in a supermarket, the probability that the barcode reader fails to read the barcode first time on any item is 0.07 , and is independent from item to item.
Calculate the probability that, from a shopping trolley containing 17 items, the reader fails to read the barcode first time on exactly 2 of the items.
Determine the probability that, from a shopping trolley containing 50 items, the reader fails to read the barcode first time on at most 5 of the items.
At another checkout in the supermarket, the probability that a faulty barcode reader fails to read the barcode first time on any item is 0.55 , and is independent from item to item.
Determine the probability that, from a shopping trolley containing 50 items, this reader fails to read the barcode first time on at least 30 of the items.
At a third checkout in the supermarket, a record is kept of \(X\), the number of times per 50 items that the barcode reader fails to read a barcode first time. An analysis of the records gives a mean of 10 and a standard deviation of 6.8.
Estimate \(p\), the probability that the barcode reader fails to read a barcode first time.
Using your estimate of \(p\) and assuming that \(X\) can be modelled by a binomial distribution, estimate the standard deviation of \(X\).
Hence comment on the assumption that \(X\) can be modelled by a binomial distribution.