6 During the winter, the probability that Barry's cat, Sylvester, chooses to stay outside all night is 0.35 , and the cat's choice is independent from night to night.
- Determine the probability that, during a period of 2 weeks ( 14 nights) in winter, Sylvester chooses to stay outside:
- on at most 7 nights;
- on at least 11 nights;
- on more than 5 nights but fewer than 10 nights.
- Calculate the probability that, during a period of \(\mathbf { 3 }\) weeks in winter, Sylvester chooses to stay outside on exactly 4 nights.
- Barry claims that, during the summer, the number of nights per week, \(S\), on which Sylvester chooses to stay outside can be modelled by a binomial distribution with \(n = 7\) and \(p = \frac { 5 } { 7 }\).
- Assuming that Barry's claim is correct, find the mean and the variance of \(S\).
- For a period of 13 weeks during the summer, the number of nights per week on which Sylvester chose to stay outside had a mean of 5 and a variance of 1.5 .
Comment on Barry's claim.
(2 marks)