2.03d Calculate conditional probability: from first principles

2. The length of time, in minutes, that a customer queues in a Post Office is a random variable, \(T\), with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } c \left( 81 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 9 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(c\) is a constant.

1. Show that the value of \(c\) is \(\frac { 1 } { 486 }\)
2. Show that the cumulative distribution function \(\mathrm { F } ( t )\) is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c c } 0 & t < 0 \\ \frac { t } { 6 } - \frac { t ^ { 3 } } { 1458 } & 0 \leqslant t \leqslant 9 \\ 1 & t > 9 \end{array} \right.$$
3. Find the probability that a customer will queue for longer than 3 minutes. A customer has been queueing for 3 minutes.
4. Find the probability that this customer will be queueing for at least 7 minutes. Three customers are selected at random.
5. Find the probability that exactly 2 of them had to queue for longer than 3 minutes.

1. David aims to catch the train to work each morning. The scheduled departure time of the train is 0830

The number of minutes after 0830 that the train departs may be modelled by the random variable \(X\). Given that \(X\) has a continuous uniform distribution over \([ \alpha , \beta ]\) and that \(\mathrm { E } ( X ) = 4\) and \(\operatorname { Var } ( X ) = 12\)

1. find the value of \(\alpha\) and the value of \(\beta\). Each morning, the probability that David oversleeps is 0.05 If David oversleeps he will be late for work. If he does not oversleep he will be in time to catch the train, but will be late for work if the train departs after 0835
2. Find the probability that David will be late for work. Given that David is late for work,
3. find the probability that he overslept.

2 Xavier, Yuri and Zara attend a sports centre for their judo club's practice sessions. The probabilities of them arriving late are, independently, \(0.3,0.4\) and 0.2 respectively.

1. Calculate the probability that for a particular practice session:
   1. all three arrive late;
   2. none of the three arrives late;
   3. only Zara arrives late.
2. Zara's friend, Wei, also attends the club's practice sessions. The probability that Wei arrives late is 0.9 when Zara arrives late, and is 0.25 when Zara does not arrive late. Calculate the probability that for a particular practice session:
   1. both Zara and Wei arrive late;
   2. either Zara or Wei, but not both, arrives late.

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 Gary and his neighbour Larry work at the same place.
On any day when Gary travels to work, he uses one of three options: his car only, a bus only or both his car and a bus. The probability that he uses his car, either on its own or with a bus, is 0.6 . The probability that he uses both his car and a bus is 0.25 .

1. Calculate the probability that, on any particular day when Gary travels to work, he:
   1. does not use his car;
   2. uses his car only;
   3. uses a bus.
2. On any day, the probability that Larry travels to work with Gary is 0.9 when Gary uses his car only, is 0.7 when Gary uses both his car and a bus, and is 0.3 when Gary uses a bus only.
   1. Calculate the probability that, on any particular day when Gary travels to work, Larry travels with him.
   2. Assuming that option choices are independent from day to day, calculate, to three decimal places, the probability that, during any particular week (5 days) when Gary travels to work every day, Larry never travels with him.

2 The number of MPs in the House of Commons was 645 at the beginning of August 2009. The genders of these MPs and the political parties to which they belonged are shown in the table.

1. One MP was selected at random for an interview. Calculate, to three decimal places, the probability that the MP was:
   1. a male Conservative;
   2. a male;
   3. a Liberal Democrat;
   4. Labour, given that the MP was female;
   5. male, given that the MP was not Labour.
2. Two female MPs were selected at random for an enquiry. Calculate, to three decimal places, the probability that both MPs were Labour.
3. Three MPs were selected at random for a committee. Calculate, to three decimal places, the probability that exactly one MP was Labour and exactly one MP was Conservative.

6 Twins Alec and Eric are members of the same local cricket club and play for the club's under 18 team. The probability that Alec is selected to play in any particular game is 0.85 .
The probability that Eric is selected to play in any particular game is 0.60 .
The probability that both Alec and Eric are selected to play in any particular game is 0.55 .

1. By using a table, or otherwise:
   1. show that the probability that neither twin is selected for a particular game is 0.10 ;
   2. find the probability that at least one of the twins is selected for a particular game;
   3. find the probability that exactly one of the twins is selected for a particular game.
2. The probability that the twins' younger brother, Cedric, is selected for a particular game is:
   0.30 given that both of the twins have been selected;
   0.75 given that exactly one of the twins has been selected;
   0.40 given that neither of the twins has been selected. Calculate the probability that, for a particular game:
   1. all three brothers are selected;
   2. at least two of the three brothers are selected.
      (6 marks)

5 Roger is an active retired lecturer. Each day after breakfast, he decides whether the weather for that day is going to be fine ( \(F\) ), dull ( \(D\) ) or wet ( \(W\) ). He then decides on only one of four activities for the day: cycling ( \(C\) ), gardening ( \(G\) ), shopping ( \(S\) ) or relaxing \(( R )\). His decisions from day to day may be assumed to be independent. The table shows Roger's probabilities for each combination of weather and activity.

1. Find the probability that, on a particular day, Roger decided:
   1. that it was going to be fine and that he would go cycling;
   2. on either gardening or shopping;
   3. to go cycling, given that he had decided that it was going to be fine;
   4. not to relax, given that he had decided that it was going to be dull;
   5. that it was going to be fine, given that he did not go cycling.
2. Calculate the probability that, on a particular Saturday and Sunday, Roger decided that it was going to be fine and decided on the same activity for both days.

2 The British and Irish Lions 2005 rugby squad contained 50 players. The nationalities and playing positions of these players are shown in the table.

1. A player was selected at random from the squad for a radio interview. Calculate the probability that the player was:
   1. a Welsh back;
   2. English;
   3. not English;
   4. Irish, given that the player was a back;
   5. a forward, given that the player was not Scottish.
2. Four players were selected at random from the squad to visit a school. Calculate the probability that all four players were English.

2 A basket in a stationery store contains a total of 400 marker and highlighter pens. Of the marker pens, some are permanent and the rest are non-permanent. The colours and types of pen are shown in the table. A pen is selected at random from the basket. Calculate the probability that it is:

1. a blue pen;
2. a marker pen;
3. a blue pen or a marker pen;
4. a green pen, given that it is a highlighter pen;
5. a non-permanent marker pen, given that it is a red pen.

5 Hugh owns a small farm.

1. He has two sows, Josie and Rosie, which he feeds at a trough in their field at 8.00 am each day. The probability that Josie is waiting at the trough at 8.00 am on any given day is 0.90 . The probability that Rosie is waiting at the trough at 8.00 am on any given day is 0.70 when Josie is waiting at the trough, but is only 0.20 when Josie is not waiting at the trough. Calculate the probability that, at 8.00 am on a given day:
   1. both sows are waiting at the trough;
   2. neither sow is waiting at the trough;
   3. at least one sow is waiting at the trough.
2. Hugh also has two cows, Daisy and Maisy. Each day at 4.00 pm , he collects them from the gate to their field and takes them to be milked. The probability, \(\mathrm { P } ( D )\), that Daisy is waiting at the gate at 4.00 pm on any given day is 0.75 .
   The probability, \(\mathrm { P } ( M )\), that Maisy is waiting at the gate at 4.00 pm on any given day is 0.60 .
   The probability that both Daisy and Maisy are waiting at the gate at 4.00 pm on any given day is 0.40 .
   1. In the table of probabilities, \(D ^ { \prime }\) and \(M ^ { \prime }\) denote the events 'not \(D\) ' and 'not \(M\) ' respectively.

4 A survey of the 640 properties on an estate was undertaken. Part of the information collected related to the number of bedrooms and the number of toilets in each property. This information is shown in the table.

1. A property on the estate is selected at random. Find, giving your answer to three decimal places, the probability that the property has:
   1. exactly 3 bedrooms;
   2. at least 2 toilets;
   3. exactly 3 bedrooms and at least 2 toilets;
   4. at most 3 bedrooms, given that it has exactly 2 toilets.
2. Use relevant answers from part (a) to show that the number of toilets is not independent of the number of bedrooms.
3. Three properties are selected at random from those on the estate which have exactly 3 bedrooms. Calculate the probability that one property has 2 toilets, one has 3 toilets and the other has at least 4 toilets. Give your answer to three decimal places.

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.

1. Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
   1. aged between 41 and 65;
   2. aged 66 or over and charged at band 2;
   3. aged between 19 and 40 and charged at most at band 1;
   4. aged 41 or over, given that the patient was charged at band 2;
   5. charged at least at band 2, given that the patient was not aged 66 or over.
2. 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]

3. The events \(A\) and \(B\) are such that $$\mathrm { P } ( A ) = 0.2 \text { and } \mathrm { P } ( A \cup B ) = 0.6$$ Find

1. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\),
2. \(\quad \mathrm { P } \left( A ^ { \prime } \cap B \right)\). Given also that events \(A\) and \(B\) are independent, find
3. \(\mathrm { P } ( B )\),
4. \(\mathrm { P } \left( A ^ { \prime } \cup B ^ { \prime } \right)\).

3. In a study of 120 pet-owners it was found that 57 owned at least one dog and of these 16 also owned at least one cat. There were 35 people in the group who didn't own any cats or dogs. As an incentive to take part in the study, one participant is chosen at random to win a year's free supply of pet food. Find the probability that the winner of this prize

1. owns a dog but does not own a cat,
2. owns a cat,
3. does not own a cat given that they do not own a dog.

1. Joel buys a box of second-hand Jazz and Blues CDs at a car boot sale.

In the box there are 30 CDs, 8 of which were recorded live. 16 of the CDs are predominantly Jazz and 13 of these were recorded in the studio. This information is shown in the following table.

1. Copy and complete the table above. Joel picks a CD at random to play first.
   Find the probability that it is
2. a Blues CD that was recorded live,
3. a Jazz CD, given that it was recorded in the studio.

4. The events \(A\) and \(B\) are such that $$\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.42 \text { and } \mathrm { P } ( A \cup B ) = 0.76$$ Find

1. \(\mathrm { P } ( A \cap B )\),
2. \(\quad \mathrm { P } \left( A ^ { \prime } \cup B \right)\),
3. \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).
4. Show that events \(A\) and \(B\) are not independent.

6. Serving against his regular opponent, a tennis player has a \(65 \%\) chance of getting his first serve in. If his first serve is in he then has a \(70 \%\) chance of winning the point but if his first serve is not in, he only has a \(45 \%\) chance of winning the point.

1. Represent this information on a tree diagram. For a point on which this player served to his regular opponent, find the probability that
2. he won the point,
3. his first serve went in given that he won the point,
4. his first serve didn't go in given that he lost the point.

3. The probability that Ajita gets up before 6.30 am in the morning is 0.7 The probability that she goes for a run in the morning is 0.35
The probability that Ajita gets up after 6.30 am and does not go for a run is 0.22
Let \(A\) represent the event that Ajita gets up before 6.30 am and \(B\) represent the event that she goes for a run in the morning. Find

1. \(\mathrm { P } ( A \cup B )\),
2. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
3. \(\mathrm { P } ( B \mid A )\).
4. State, with a reason, whether or not events \(A\) and \(B\) are independent.

2. Events \(A\) and \(B\) are independent. Given also that $$\mathrm { P } ( A ) = \frac { 3 } { 4 } \quad \text { and } \quad \mathrm { P } \left( A \cap B ^ { \prime } \right) = \frac { 1 } { 4 }$$ Find

1. \(\mathrm { P } ( A \cap B )\),
2. \(\mathrm { P } ( B )\),
3. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\).

6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.

1. Show that the probability that the first two contestants to be eliminated are both male is \(\frac { 2 } { 15 }\).
2. Find the probability that more females than males are eliminated in the first three rounds of the game.
3. Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
   (3 marks)

4 A discrete random variable \(X\) has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 3 x } { 40 } & x = 1,2,3,4 \\ \frac { x } { 20 } & x = 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Calculate \(\mathrm { E } ( X )\).
2. Show that:
   1. \(\quad \mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 20 }\);
      (2 marks)
   2. \(\operatorname { Var } \left( \frac { 1 } { X } \right) = \frac { 7 } { 160 }\).
3. The discrete random variable \(Y\) is such that \(Y = \frac { 40 } { X }\). Calculate:
   1. \(\mathrm { P } ( Y < 20 )\);
   2. \(\mathrm { P } ( X < 4 \mid Y < 20 )\).

7 A continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } ( 4 - x ) & 1 \leqslant x \leqslant 3 \\ \frac { 1 } { 6 } & 3 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Draw the graph of f on the grid on page 6 .
2. Prove that the mean of \(X\) is \(2 \frac { 5 } { 9 }\).
3. Calculate the exact value of:
   1. \(\mathrm { P } ( X > 2.5 )\);
   2. \(\mathrm { P } ( 1.5 < X < 4.5 )\);
   3. \(\mathrm { P } ( X > 2.5\) and \(1.5 < X < 4.5 )\);
   4. \(\mathrm { P } ( X > 2.5 \mid 1.5 < X < 4.5 )\). \includegraphics[max width=\textwidth, alt={}, center]{bc21c177-6cd8-4c79-8782-d17f0238ce17-6_1340_1363_317_383}

2 A hotel chain has hotels in three types of location: city, coastal and country. The percentages of the chain's reservations for each of these locations are 30,55 and 15 respectively. Each of the chain's hotels offers three types of reservation: Bed \& Breakfast, Half Board and Full Board. The percentages of these types of reservation for each of the three types of location are shown in the table. For example, 80 per cent of reservations for hotels in city locations are for Bed \& Breakfast.

1. For a reservation selected at random:
   1. show that the probability that it is for Bed \& Breakfast is 0.34 ;
   2. calculate the probability that it is for Half Board in a hotel in a coastal location;
   3. calculate the probability that it is for a hotel in a coastal location, given that it is for Half Board.
2. A random sample of 3 reservations for Half Board is selected. Calculate the probability that these 3 reservations are for hotels in different types of location.

3 An IT help desk has three telephone stations: Alpha, Beta and Gamma. Each of these stations deals only with telephone enquiries. The probability that an enquiry is received at Alpha is 0.60 .
The probability that an enquiry is received at Beta is 0.25 .
The probability that an enquiry is received at Gamma is 0.15 . Each enquiry is resolved at the station that receives the enquiry. The percentages of enquiries resolved within various times at each station are shown in the table. For example:
80 per cent of enquiries received at Alpha are resolved within 24 hours;
25 per cent of enquiries received at Alpha take between 1 hour and 24 hours to resolve.

1. Find the probability that an enquiry, selected at random, is:
   1. resolved at Gamma;
   2. resolved at Alpha within 1 hour;
   3. resolved within 24 hours;
   4. received at Beta, given that it is resolved within 24 hours.
2. A random sample of 3 enquiries was selected. Given that all 3 enquiries were resolved within 24 hours, calculate the probability that they were all received at:
   1. Beta;
   2. the same station.