Basic probability calculation

9 A fair six-sided die has its faces numbered 1, 3, 4, 5, 6 and 7. The die is rolled once.
\(A\) is the event that the die shows an even number.
\(B\) is the event that the die shows a prime number.

1. Write down the value of \(\mathrm { p } ( A )\).
2. Write down the value of \(\mathrm { p } ( B )\).
3. Write down the value of \(\mathrm { p } ( A\) or \(B )\). The die is rolled again.
4. Calculate the probability that the sum of the scores from the two rolls is even.

3 The probability that Chipping FC win a league football match is \(\mathrm { P } ( W ) = 0.4\).

1. Calculate the probability that Chipping FC fail to win each of their next two league football matches. The probability that Chipping FC lose a league football match is \(\mathrm { P } ( L ) = 0.3\).
2. Explain why \(\mathrm { P } ( W ) + \mathrm { P } ( L ) \neq 1\).

1. Jake and Kamil are sometimes late for school.

The events \(J\) and \(K\) are defined as follows
\(J =\) the event that Jake is late for school
\(K =\) the event that Kamil is late for school
\(\mathrm { P } ( J ) = 0.25 , \mathrm { P } ( J \cap K ) = 0.15\) and \(\mathrm { P } \left( J ^ { \prime } \cap K ^ { \prime } \right) = 0.7\) On a randomly selected day, find the probability that

1. at least one of Jake or Kamil are late for school,
2. Kamil is late for school. Given that Jake is late for school,
3. find the probability that Kamil is late. The teacher suspects that Jake being late for school and Kamil being late for school are linked in some way.
4. Determine whether or not \(J\) and \(K\) are statistically independent.
5. Comment on the teacher's suspicion in the light of your calculation in (d).

5 A health club has a number of facilities which include a gym and a sauna. Andrew and his wife, Heidi, visit the health club together on Tuesday evenings. On any visit, Andrew uses either the gym or the sauna or both, but no other facilities. The probability that he uses the gym, \(\mathrm { P } ( G )\), is 0.70 . The probability that he uses the sauna, \(\mathrm { P } ( S )\), is 0.55 . The probability that he uses both the gym and the sauna is 0.25 .

1. Calculate the probability that, on a particular visit:
   1. he does not use the gym;
   2. he uses the gym but not the sauna;
   3. he uses either the gym or the sauna but not both.
2. Assuming that Andrew's decision on what facility to use is independent from visit to visit, calculate the probability that, during a month in which there are exactly four Tuesdays, he does not use the gym.
3. The probability that Heidi uses the gym when Andrew uses the gym is 0.6 , but is only 0.1 when he does not use the gym. Calculate the probability that, on a particular visit, Heidi uses the gym.
4. On any visit, Heidi uses exactly one of the club's facilities. The probability that she uses the sauna is 0.35 .
   Calculate the probability that, on a particular visit, she uses a facility other than the gym or the sauna.

4 Alf and Mabel are members of a bowls club and sometimes attend the club's social events. The probability, \(\mathrm { P } ( A )\), that Alf attends a social event is 0.70 .
The probability, \(\mathrm { P } ( M )\), that Mabel attends a social event is 0.55 .
The probability, \(\mathrm { P } ( A \cap M )\), that both Alf and Mabel attend the same social event is 0.45 .

1. Find the probability that:
   1. either Alf or Mabel or both attend a particular social event;
   2. either Alf or Mabel but not both attend a particular social event.
2. Give a numerical justification for the following statement.
   "Events \(A\) and \(M\) are not independent."
3. Ben and Nora are also members of the bowls club and sometimes attend the club's social events. The probability, \(\mathrm { P } ( B )\), that Ben attends a social event is 0.85 .
   The probability, \(\mathrm { P } ( N )\), that Nora attends a social event is 0.65 .
   The attendance of each of Ben and Nora at a social event is independent of the attendance of all other members. Find the probability that:
   1. all four named members attend a particular social event;
   2. none of the four named members attend a particular social event.

1 A council claims that 80 per cent of households are generally satisfied with the services it provides. A random sample of 250 households shows that 209 are generally satisfied with the council's provision of services.

1. Construct an approximate \(95 \%\) confidence interval for the proportion of households that are generally satisfied with the council's provision of services.
2. Hence comment on the council's claim.