7 The proportion of passengers who use senior citizen bus passes to travel into a particular town on 'Park \& Ride' buses between 9.30 am and 11.30 am on weekdays is 0.45 . It is proposed that, when there are \(n\) passengers on a bus, a suitable model for the number of passengers using senior citizen bus passes is the distribution \(\mathrm { B } ( n , 0.45 )\).

1. Assuming that this model applies to the 10.30 am weekday 'Park \& Ride' bus into the town:
   1. calculate the probability that, when there are \(\mathbf { 1 6 }\) passengers, exactly 3 of them are using senior citizen bus passes;
   2. determine the probability that, when there are \(\mathbf { 2 5 }\) passengers, fewer than 10 of them are using senior citizen bus passes;
   3. determine the probability that, when there are \(\mathbf { 4 0 }\) passengers, at least 15 but at most 20 of them are using senior citizen bus passes;
   4. calculate the mean and the variance for the number of passengers using senior citizen bus passes when there are \(\mathbf { 5 0 }\) passengers.
2. 1. Give a reason why the proposed model may not be suitable.
   2. Give a different reason why the proposed model would not be suitable for the number of passengers using senior citizen bus passes to travel into the town on the 7.15 am weekday 'Park \& Ride' bus.