3 The table shows, for a random sample of 500 patients attending a dental surgery, the patients' ages, in years, and the NHS charge bands for the patients' courses of treatment. Band 0 denotes the least expensive charge band and band 3 denotes the most expensive charge band.
| \multirow{2}{*}{} | Charge band for course of treatment | |
| | Band 0 | Band 1 | Band 2 | Band 3 | Total |
| \multirow{4}{*}{Age of patient (years)} | Under 19 | 32 | 43 | 5 | 0 | 80 |
| Between 19 and 40 | 17 | 62 | 22 | 3 | 104 |
| Between 41 and 65 | 28 | 82 | 35 | 31 | 176 |
| 66 or over | 13 | 53 | 68 | 6 | 140 |
| Total | 90 | 240 | 130 | 40 | 500 |
- Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
- aged between 41 and 65;
- aged 66 or over and charged at band 2;
- aged between 19 and 40 and charged at most at band 1;
- aged 41 or over, given that the patient was charged at band 2;
- charged at least at band 2, given that the patient was not aged 66 or over.
- Four patients at this dental surgery, not included in the above 500 patients, are selected at random.
Estimate, to three significant figures, the probability that two of these four patients are aged between 41 and 65 and are not charged at band 0 , and the other two patients are aged 66 or over and are charged at either band 1 or band 2.
[0pt]
[5 marks]