2.03a Mutually exclusive and independent events

4.A bag contains 64 coloured beads.There are \(r\) red beads,\(y\) yellow beads and 1 green bead and \(r + y + 1 = 64\) Two beads are selected at random,one at a time without replacement.

1. Find the probability that the green bead is one of the beads selected. The probability that both of the beads are red is \(\frac { 5 } { 84 }\)
2. Show that \(r\) satisfies the equation \(r ^ { 2 } - r - 240 = 0\)
3. Hence show that the only possible value of \(r\) is 16
4. Given that at least one of the beads is red,find the probability that they are both red.

4. Explain what you understand by

1. a sample space,
2. an event. Two events \(A\) and \(B\) are independent, such that \(\mathrm { P } ( A ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( B ) = \frac { 1 } { 4 }\).
   Find
3. \(\mathrm { P } ( A \cap B )\),
4. \(\mathrm { P } ( A B )\),
5. \(\mathrm { P } ( A \cup B )\).

5. The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( B ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( A \cap B ) = \frac { 1 } { 4 }\).

1. Using the space below, represent these probabilities in a Venn diagram. Hence, or otherwise, find
2. \(\mathrm { P } ( A \cup B )\),
3. \(\mathrm { P } \left( \begin{array} { l l } A & B ^ { \prime } \end{array} \right)\)

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.

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.

1 A large bookcase contains two types of book: hardback and paperback. The number of books of each type in each of four subject categories is shown in the table.

1. A book is selected at random from the bookcase. Calculate the probability that the book is:
   1. a paperback;
   2. not science fiction;
   3. science fiction or a hardback;
   4. a thriller, given that it is a paperback.
2. Three books are selected at random, without replacement, from the bookcase. Calculate, to three decimal places, the probability that one is crime, one is romance and one is science fiction.

5

1. Emma visits her local supermarket every Thursday to do her weekly shopping. The event that she buys orange juice is denoted by \(J\), and the event that she buys bottled water is denoted by \(W\). At each visit, Emma may buy neither, or one, or both of these items.
   1. Complete the table of probabilities, printed below, for these events, where \(J ^ { \prime }\) and \(W ^ { \prime }\) denote the events 'not \(J\) ' and 'not \(W ^ { \prime }\) respectively.
   2. Hence, or otherwise, find the probability that, on any given Thursday, Emma buys either orange juice or bottled water but not both.
   3. Show that:
      (A) the events \(J\) and \(W\) are not mutually exclusive;
      (B) the events \(J\) and \(W\) are not independent.
2. Rhys visits the supermarket every Saturday to do his weekly shopping. Items that he may buy are milk, cheese and yogurt. The probability, \(\mathrm { P } ( M )\), that he buys milk on any given Saturday is 0.85 .
   The probability, \(\mathrm { P } ( C )\), that he buys cheese on any given Saturday is 0.60 .
   The probability, \(\mathrm { P } ( Y )\), that he buys yogurt on any given Saturday is 0.55 .
   The events \(M , C\) and \(Y\) may be assumed to be independent. Calculate the probability that, on any given Saturday, Rhys buys:
   1. none of the 3 items;
   2. exactly 2 of the 3 items.

      \cline { 2 - 4 } \multicolumn{1}{c|}{}	\(\boldsymbol { J }\)	\(\boldsymbol { J } ^ { \prime }\)	Total
      \(\boldsymbol { W }\)			0.65
      \(\boldsymbol { W } ^ { \prime }\)	0.15
      Total		0.30	1.00

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 An auction house offers items of jewellery for sale at its public auctions. Each item has a reserve price which is less than the lower price estimate which, in turn, is less than the upper price estimate. The outcome for any item is independent of the outcomes for all other items. The auction house has found, from past records, the following probabilities for the outcomes of items of jewellery offered for sale.

5 Alison is a member of a tenpin bowling club which meets at a bowling alley on Wednesday and Thursday evenings. The probability that she bowls on a Wednesday evening is 0.90 . Independently, the probability that she bowls on a Thursday evening is 0.95 .

1. Calculate the probability that, during a particular week, Alison bowls on:
   1. two evenings;
   2. exactly one evening.
2. David, a friend of Alison, is a member of the same club. The probability that he bowls on a Wednesday evening, given that Alison bowls on that evening, is 0.80 . The probability that he bowls on a Wednesday evening, given that Alison does not bowl on that evening, is 0.15 . The probability that he bowls on a Thursday evening, given that Alison bowls on that evening, is 1 . The probability that he bowls on a Thursday evening, given that Alison does not bowl on that evening, is 0 . Calculate the probability that, during a particular week:
   1. Alison and David bowl on a Wednesday evening;
   2. Alison and David bowl on both evenings;
   3. Alison, but not David, bowls on a Thursday evening;
   4. neither bowls on either evening.

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.

2. The events \(A\) and \(B\) are independent. Given that \(\mathrm { P } ( A ) = 0.4\) and \(\mathrm { P } ( A \cap B ) = 0.12\), find

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

2. Given that \(\mathrm { P } ( A ) = \frac { 3 } { 5 } , \mathrm { P } ( B ) = \frac { 5 } { 8 } , \mathrm { P } ( A \cap B ) = \frac { 7 } { 20 } , \mathrm { P } ( A \cup C ) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid A ) = \frac { 1 } { 3 }\),

1. determine whether \(A\) and \(B\) are independent events.
2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
3. Find \(\mathrm { P } \left( ( A \cap C ) ^ { \prime } \right)\).
4. Find \(\mathrm { P } ( A \mid C )\).

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. \(\mathrm { P } \left( A ^ { \prime } \cup B ^ { \prime } \right)\).

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.

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.

5. A College employs 75 teachers, of whom 47 are full-time and the rest are part-time. Of the 39 male teachers at the College, 26 are full-time.

1. Represent this information on a Venn diagram.
2. One teacher is selected at random to be interviewed by an inspector. Find the probability that the teacher chosen
   1. works full-time and is female,
   2. works part-time, given that he is male.
3. Three teachers are selected at random to be observed by an inspector during one day. Find correct to 3 significant figures the probability that
   1. all three teachers chosen work full-time,
   2. at least one of the three teachers chosen is female.

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. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\).

2 On a rail route between two stations, A and \(\mathrm { B } , 90 \%\) of trains leave A on time and \(10 \%\) of trains leave A late. Of those trains that leave A on time, \(15 \%\) arrive at B early, \(75 \%\) arrive on time and \(10 \%\) arrive late. Of those trains that leave A late, \(35 \%\) arrive at B on time and \(65 \%\) arrive late.

1. Represent this information by a fully-labelled tree diagram.
2. Hence, or otherwise, calculate the probability that a train:
   1. arrives at B early or on time;
   2. left A on time, given that it arrived at B on time;
   3. left A late, given that it was not late in arriving at B .
3. Two trains arrive late at B. Assuming that their journey times are independent, calculate the probability that exactly one train left A on time.

3 An investigation was carried out into the type of vehicle being driven when its driver was caught speeding. The investigation was restricted to drivers who were caught speeding when driving vehicles with at least 4 wheels. An analysis of the results showed that \(65 \%\) were driving cars ( C ), \(20 \%\) were driving vans (V) and 15\% were driving lorries (L). Of those driving cars, \(30 \%\) were caught by fixed speed cameras (F), 55\% were caught by mobile speed cameras (M) and 15\% were caught by average speed cameras (A). Of those driving vans, \(35 \%\) were caught by fixed speed cameras (F), \(45 \%\) were caught by mobile speed cameras (M) and 20\% were caught by average speed cameras (A). Of those driving lorries, \(10 \%\) were caught by fixed speed cameras \(( \mathrm { F } )\), \(65 \%\) were caught by mobile speed cameras (M) and \(25 \%\) were caught by average speed cameras (A).

1. Represent this information by a tree diagram on which are shown labels and percentages or probabilities.
2. Hence, or otherwise, calculate the probability that a driver, selected at random from those caught speeding:
   1. was driving either a car or a lorry and was caught by a mobile speed camera;
   2. was driving a lorry, given that the driver was caught by an average speed camera;
   3. was not caught by a fixed speed camera, given that the driver was not driving a car.
      [0pt] [8 marks]
3. Three drivers were selected at random from those caught speeding by fixed speed cameras. Calculate the probability that they were driving three different types of vehicle.
   [0pt] [4 marks]