2.03b Probability diagrams: tree, Venn, sample space

The Venn diagram shows three events \(A , B\) and \(C\), where \(p , q , r , s\) and \(t\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{319667e7-3f8b-4a33-8fc5-ef72154d1421-10_647_972_306_488}
(b) Find the value of \(r\).
(c) Hence write down the value of \(s\) and the value of \(t\).
(d) State, giving a reason, whether or not the events \(A\) and \(B\) are independent.
(e) Find \(\mathrm { P } ( B \mid A \cup C )\). \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } ( C ) = 0.25\) and the events \(B\) and \(C\) are independent.
(a) Find the value of \(p\) and the value of \(q\).

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.

5. A keep-fit enthusiast swims, runs or cycles each day with probabilities \(0.2,0.3\) and 0.5 respectively. If he swims he then spends time in the sauna with probability 0.35 . The probabilities that he spends time in the sauna after running or cycling are 0.2 and 0.45 respectively.

1. Represent this information on a tree diagram.
2. Find the probability that on any particular day he uses the sauna.
3. Given that he uses the sauna one day, find the probability that he had been swimming.
4. Given that he did not use the sauna one day, find the probability that he had been swimming.

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. \(\mathrm { P } \left( \begin{array} { l l } A & B ^ { \prime } \end{array} \right)\)

1. At a cafe, customers ordering hot drinks order either tea or coffee.

Of all customers ordering hot drinks, \(80 \%\) order tea and \(20 \%\) order coffee. Of those who order tea, \(35 \%\) take sugar and of those who order coffee \(60 \%\) take sugar.

1. A random sample of 12 customers ordering hot drinks is selected. Find the probability that fewer than 3 of these customers order coffee.
2. 1. A randomly selected customer who orders a hot drink is chosen. Show that the probability that the customer takes sugar is 0.4
   2. Write down the distribution for the number of customers who take sugar from a random sample of \(n\) customers ordering hot drinks.
3. A random sample of 10 customers ordering hot drinks is selected.
   1. Find the probability that exactly 4 of these 10 customers take sugar.
   2. Given that at least 3 of these 10 customers take sugar, find the probability that no more than 6 of these 10 customers take sugar.
4. In a random sample of 150 customers ordering hot drinks, find, using a suitable approximation, the probability that at least half of them take sugar.

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.

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.

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

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.

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]

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

6. The individual letters of the word STATISTICAL are written on 11 cards which are then shuffled. One card is selected at random. Find the probability that it is

1. a vowel,
2. a T, given that it is a consonant. The 11 cards are then shuffled again and the top three are turned over. Find the probability that
3. all three of the cards have a T on them,
4. at least two of the cards show a vowel.

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.
   

6. A software company sets exams for programmers who wish to qualify to use their packages. Past records show that \(55 \%\) of candidates taking the exam for the first time will pass, \(60 \%\) of those taking it for the second time will pass, but only \(40 \%\) of those taking the exam for the third time will pass. Candidates are not allowed to sit the exam more than three times. A programmer decides to keep taking the exam until he passes or is allowed no further attempts. Find the probability that he will

1. pass the exam on his second attempt,
2. pass the exam. Another programmer already has the qualification.
3. Find, correct to 3 significant figures, the probability that she passed first time. At a particular sitting of the exam there are 400 candidates.
   The ratio of those sitting the exam for the first time to those sitting it for the second time to those sitting it for the third time is \(5 : 3 : 2\)
4. How many of the 400 candidates would be expected to pass?

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.