5 An analysis of the number of vehicles registered by each household within a city resulted in the following information.
| Number of vehicles registered | 0 | 1 | 2 | \(\geqslant 3\) |
| Percentage of households | 18 | 47 | 25 | 10 |
- A random sample of 30 households within the city is selected.
Use a binomial distribution with \(n = 30\), together with relevant information from the table in each case, to find the probability that the sample contains:
- exactly 3 households with no registered vehicles;
- at most 5 households with three or more registered vehicles;
- more than 10 households with at least two registered vehicles;
- more than 5 households but fewer than 10 households with exactly two registered vehicles.
- If a random sample of \(\mathbf { 1 5 0 }\) households within the city were to be selected, estimate the mean and the variance for the number of households in the sample that would have either one or two registered vehicles.
[0pt]
[2 marks]
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