2 An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.

1. Three days are chosen at random. Find the probability that the pollution level is
   (A) low on all 3 days,
   (B) low on at least one day,
   (C) low on one day, medium on another day, and high on the other day.
2. Ten days are chosen at random. Find the probability that
   (A) there are no days when the pollution level is high,
   (B) there is exactly one day when the pollution level is high. The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
3. Carry out a test at the \(5 \%\) level to determine if there is evidence to suggest that she is correct. Use hypotheses \(\mathrm { H } _ { 0 } : p = 0.5 , \mathrm { H } _ { 1 } : p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(\mathrm { H } _ { 1 }\) has this form.