2.03b Probability diagrams: tree, Venn, sample space

7 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.

1. Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
2. Find the probability that
   (A) all three of these trains are on time,
   (B) just one of these three trains is on time,
   (C) the 1200 train is on time.
3. Given that the 1200 train is on time, find the probability that the 1000 train is also on time. 3
4. Write any calculations on page 5. \includegraphics[max width=\textwidth, alt={}, center]{091d6f43-ad01-4849-9f3c-3e58349aa169-4_2276_1490_324_363}

7 Jenny has six darts. She throws darts, one at a time, aiming each at the bull's-eye. The probability that she hits the bull's-eye with her first dart is 0.1 . For any subsequent throw, the probability of hitting the bull's-eye is 0.2 if the previous dart hit the bull's-eye and 0.05 otherwise.

1. Illustrate the possible outcomes for her first, second and third darts on a probability tree diagram.
2. Find the probability that
   (A) she hits the bull's-eye with at least one of her first three darts,
   (B) she hits the bull's-eye with exactly one of her first three darts.
3. Given that she hits the bull's-eye with at least one of her first three darts, find the probability that she hits the bull's-eye with exactly one of them. Jenny decides that, if she hits the bull's-eye with any of her first three darts, she will stop after throwing three darts. Otherwise she will throw all six darts.
4. Find the probability that she hits the bull's-eye three times in total.

2 A survey is being carried out into the sports viewing habits of people in a particular area. As part of the survey, 250 people are asked which of the following sports they have watched on television in the past month.

• Football
• Cycling
• Rugby

The numbers of people who have watched these sports are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-2_723_917_1183_575} One of the people is selected at random.

1. Find the probability that this person has in the past month
   (A) watched cycling but not football,
   (B) watched either one or two of the three sports.
2. Given that this person has watched cycling, find the probability that this person has not watched football.

1 For the events \(A\) and \(B\) it is given that $$\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.3 \text { and } \mathrm { P } ( A \text { or } B \text { but not both } ) = 0.4 \text {. }$$

1. Find \(\mathrm { P } ( A \cap B )\).
2. Find \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\).
3. State, giving a reason, whether \(A\) and \(B\) are independent.

3 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup B ) = 0.8\).

1. Find \(\mathrm { P } ( A \cap B )\).
2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
3. Find \(\mathrm { P } ( A \mid B )\). Events \(A\) and \(B\) are as above and a third event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1 , \mathrm { P } ( A \cap B \cap C ) = 0.05\), \(\mathrm { P } ( A \cap C ) = \mathrm { P } ( B \cap C )\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 3 \mathrm { P } \left( A ^ { \prime } \cap B \cap C ^ { \prime } \right)\).
4. Find \(\mathrm { P } ( C )\).

1 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a \(\pounds 10\) prize, 20 of them have a \(\pounds 100\) prize, one of them has a \(\pounds 5000\) prize and all of the rest have no prize. This information is summarised in the frequency table below.

1. Find the mean and standard deviation of the prize money per ticket.
2. I buy two of these tickets at random. Find the probability that I win either two \(\pounds 10\) prizes or two \(\pounds 100\) prizes.

5 Each day Anna drives to work.

• \(R\) is the event that it is raining.
• \(L\) is the event that Anna arrives at work late.

You are given that \(\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25\) and \(\mathrm { P } ( R \cap L ) = 0.2\).

1. Determine whether the events \(R\) and \(L\) are independent.
2. Draw a Venn diagram showing the events \(R\) and \(L\). Fill in the probability corresponding to each of the four regions of your diagram.
3. Find \(\mathrm { P } ( L \mid R )\). State what this probability represents.

2 In a hockey league, each team plays every other team 3 times. The probabilities that Team A wins, draws and loses to Team B are given below.

• \(\mathrm { P } (\) Wins \() = 0.5\)
• \(\mathrm { P } (\) Draws \() = 0.3\)
• \(\mathrm { P } (\) Loses \() = 0.2\)

The outcomes of the 3 matches are independent.

1. Find the probability that Team A does not lose in any of the 3 matches.
2. Find the probability that Team A either wins all 3 matches or draws all 3 matches or loses all 3 matches.
3. Find the probability that, in the 3 matches, exactly two of the outcomes, 'Wins', 'Draws' and 'Loses' occur for Team A.

5 Measurements of sunshine and rainfall are made each day at a particular weather station. For a randomly chosen day,

• \(R\) is the event that at least 1 mm of rainfall is recorded,
• \(S\) is the event that at least 1 hour of sunshine is recorded.

You are given that \(\mathrm { P } ( R ) = 0.28 , \mathrm { P } ( S ) = 0.87\) and \(\mathrm { P } ( R \cup S ) = 0.94\).

1. Find \(\mathrm { P } ( R \cap S )\).
2. Draw a Venn diagram showing the events \(R\) and \(S\), and fill in the probability corresponding to each of the four regions of your diagram.
3. Find \(\mathrm { P } ( R \mid S )\) and state what this probability represents in this context.

5 Each morning I reach into my box of tea bags and, without looking, randomly choose a bag. The bags are manufactured in pairs, which can be separated along a perforated line. So when I choose a bag it might be attached to another, in which case I have to separate them and return the other bag to the box. Alternatively, it might be a single bag, having been separated on an earlier day. I only use one tea bag per day, and the box always gets thoroughly shaken during the day as things are moved around in the kitchen. You are to simulate this process, starting with 5 double bags and 0 single bags in the box. You are to use single-digit random numbers in your simulation.

1. On day 2 there will be 4 double bags and 1 single bag in the box, 9 bags in total. Give a rule for simulating whether I choose a single bag or a double bag, assuming that I am equally likely to choose any of the 9 bags. Use single-digit random numbers in your simulation rule.
2. On day 3 there will either be 4 double bags or 3 double bags and 2 single bags in the box. Give a rule for simulating what sort of bag I choose in the second of these cases. Use single-digit random numbers in your simulation rule.
3. Using the random digits in your answer book, simulate what happens on days 2,3 and 4 , briefly explaining your simulations. Give an estimate of the probability that I choose a single bag on day 5 .
4. Using the random digits in your answer book, carry out 4 more simulations and record the results.
5. Using your 5 simulations, estimate the probability that I choose a single bag on day 5 .
   [0pt] [Question 6 is printed overleaf.]

2 Honor either has coffee or tea at breakfast. On one third of days she chooses coffee, otherwise she has tea. She can never remember what she had the day before.

1. Construct a simulation rule, using one-digit random numbers, to model Honor's choices of breakfast drink.
2. Using the one-digit random numbers in your answer book, simulate Honor's choice of breakfast drink for 10 days. Honor also has either coffee or tea at the end of her evening meal, but she does remember what she had for breakfast, and her choice depends on it. If she had coffee at breakfast then the probability of her having coffee again is 0.55 . If she had tea for breakfast, then the probability of her having tea again is 0.15 .
3. Construct a simulation rule, using two-digit random numbers, to model Honor's choice of evening drink given that she had coffee at breakfast. Construct a simulation rule, using two-digit random numbers, to model Honor's choice of evening drink given that she had tea at breakfast.
4. Using your breakfast simulation from part (ii), and the two-digit random numbers in your answer book, simulate Honor's choice of evening drink for 10 days.
5. Use your results from parts (ii) and (iv) to estimate the proportion of Honor's drinks, breakfast and evening meal combined, which are coffee. \section*{Question 3 begins on page 4}

6 Adrian and Kleo like to go out for meals, sometimes to a French restaurant, and sometimes to a Greek restaurant. If their last meal out was at the French restaurant, then the probability of their next meal out being at the Greek restaurant is 0.7 , whilst the probability of it being at the French restaurant is 0.3 . If their last meal out was at the Greek restaurant, then the probability of their next meal out being at the French restaurant is 0.6 , whilst the probability of it being at the Greek restaurant is 0.4 .

1. Construct two simulation rules, each using single-digit random numbers, to model their choices of where to eat.
2. Their last meal out was at the Greek restaurant. Use the random digits printed in your answer book to simulate their choices for the next 10 of their meals out. Hence estimate the proportion of their meals out which are at the French restaurant, and the proportion which are at the Greek restaurant. Adrian and Kleo find a Hungarian restaurant which they like. The probabilities of where they eat next are now given in the following table.

   \backslashbox{last meal out}{next meal out}	French	Greek	Hungarian
   French	\(\frac { 1 } { 5 }\)	\(\frac { 3 } { 5 }\)	\(\frac { 1 } { 5 }\)
   Greek	\(\frac { 1 } { 2 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 1 } { 5 }\)
   Hungarian	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 3 }\)

3. Construct simulation rules, each using single-digit random numbers, to model this new situation.
4. Their last meal out was at the Greek restaurant. Use the random digits printed in your answer book to simulate their choices for the next 10 of their meals out. Hence estimate the proportion of their meals out which are at each restaurant. \section*{END OF QUESTION PAPER}

1 Pierre knows that, if he gambles, he will lose money in the long run. Nicolas tries to convince him that this is not the case. Pierre stakes a sum of money in a casino game. If he wins then he gets back his stake plus the same amount again. If he loses then he loses his stake. Nicolas says that Pierre can guarantee to win by repeatedly playing the game, even though the probability of winning an individual game is less than 0.5 . His idea is that Pierre should bet in the first game with a stake of \(\pounds 100\). If he wins then he stops, as he will have won \(\pounds 100\). If he loses then he plays again with a stake of \(\pounds 200\). If he wins then he has lost \(\pounds 100\) and won \(\pounds 200\). This gives a total gain of \(\pounds 100\), and he stops. If he loses then he plays again with a stake of \(\pounds 400\). If he wins this time he has lost \(\pounds 100\) and \(\pounds 200\) and won \(\pounds 400\). This gives a total gain of \(\pounds 100\), and he stops. Nicolas's advice is that Pierre simply has to continue in this way, doubling his stake every time that he loses, until he eventually wins. Nicolas says that this guarantees that Pierre will win \(\pounds 100\). You are to simulate what might happen if Pierre tries this strategy in a casino game in which the probability of him winning an individual game is 0.4 , and in which he has \(\pounds 1000\) available.

1. Give an efficient rule for using 1-digit random numbers to simulate the outcomes of individual games, given that the probability of Pierre winning an individual game is 0.4 .
2. Explain why at most three random digits are needed for one simulation of Nicolas's strategy, given that Pierre is starting with \(\pounds 1000\).
3. Simulate five applications of Nicolas's strategy, using the five sets of three 1-digit random numbers in your answer book.
4. Summarise the results of your simulations, giving your mean result.

12 Anika and Beth are playing a game which consists of several points.

• The probability that Anika will win any point is 0.7 .
• The probability that Beth will win any point is 0.3 .
• The outcome of each point is independent of the outcome of every other point.

The first player to win two points wins the game.

1. Write down the probability that the game consists of more than three points.
2. Complete the probability tree diagram in the Printed Answer Booklet showing all the possibilities for the game.
3. Determine the probability that Beth wins the game.
4. Determine the probability that the game consists of exactly three points.
5. Given that Beth wins the game, determine the probability that the game consists of exactly three points.

3. In an after-school club, students can choose to take part in Art, Music, both or neither. There are 45 students that attend the after-school club. Of these

• 25 students take part in Art
• 12 students take part in both Art and Music
• the number of students that take part in Music is \(x\)
  1. Find the range of possible values of \(x\)

One of the 45 students is selected at random.
Event \(A\) is the event that the student selected takes part in Art.
Event \(M\) is the event that the student selected takes part in Music.

Determine whether or not it is possible for the events \(A\) and \(M\) to be independent.

1. \includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-02_399_743_248_662} The Venn diagram, where \(p\) is a probability, shows the 3 events \(A , B\) and \(C\) with their associated probabilities.

1. Find the value of \(p\).
2. Write down a pair of mutually exclusive events from \(A , B\) and \(C\).

1. Three bags, \(A , B\) and \(C\), each contain 1 red marble and some green marbles.

Bag \(A\) contains 1 red marble and 9 green marbles only
Bag \(B\) contains 1 red marble and 4 green marbles only
Bag \(C\) contains 1 red marble and 2 green marbles only
Sasha selects at random one marble from bag \(A\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag \(B\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag \(C\).

1. Draw a tree diagram to represent this information.
2. Find the probability that Sasha selects 3 green marbles.
3. Find the probability that Sasha selects at least 1 marble of each colour.
4. Given that Sasha selects a red marble, find the probability that he selects it from bag \(B\).

1. A company has 1825 employees.

The employees are classified as professional, skilled or elementary.
The following table shows

• the number of employees in each classification
• the two areas, \(A\) or \(B\), where the employees live

An employee is chosen at random.
Find the probability that this employee

1. is skilled,
2. lives in area \(B\) and is not a professional. Some classifications of employees are more likely to work from home.
   • \(65 \%\) of professional employees in both area \(A\) and area \(B\) work from home
   • \(40 \%\) of skilled employees in both area \(A\) and area \(B\) work from home
   • \(5 \%\) of elementary employees in both area \(A\) and area \(B\) work from home
   • Event \(F\) is that the employee is a professional
   • Event \(H\) is that the employee works from home
   • Event \(R\) is that the employee is from area \(A\)
   • Using this information, complete the Venn diagram on the opposite page.
   • Find \(\mathrm { P } \left( R ^ { \prime } \cap F \right)\)
   • Find \(\mathrm { P } \left( [ H \cup R ] ^ { \prime } \right)\)
   • Find \(\mathrm { P } ( F \mid H )\)
   \includegraphics[max width=\textwidth, alt={}]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-13_872_1020_294_525}
   Turn over for a spare diagram if you need to redraw your Venn diagram. Only use this diagram if you need to redraw your Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-15_872_1017_392_525}

The Venn diagram, where \(p\) and \(q\) are probabilities, shows the three events \(A , B\) and \(C\) and their associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{a067577e-e2a6-440b-9d22-d558fade15f0-02_745_935_347_566}

1. Find \(\mathrm { P } ( A )\) The events \(B\) and \(C\) are independent.
2. Find the value of \(p\) and the value of \(q\)
3. Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\)

1. The Venn diagram, where \(p , q\) and \(r\) are probabilities, shows the events \(A , B , C\) and \(D\) and associated probabilities.

\begin{figure}[h] \end{figure}

1. State any pair of mutually exclusive events from \(A\), \(B\), \(C\) and \(D\) The events \(B\) and \(C\) are independent.
2. Find the value of \(p\)
3. Find the greatest possible value of \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\) Given that \(\mathrm { P } \left( B \mid A ^ { \prime } \right) = 0.5\)
4. find the value of \(q\) and the value of \(r\)
5. Find \(\mathrm { P } \left( [ A \cup B ] ^ { \prime } \cap C \right)\)
6. Use set notation to write an expression for the event with probability \(p\)

The Venn diagram shows the probabilities associated with four events, \(A , B , C\) and \(D\) \includegraphics[max width=\textwidth, alt={}, center]{2b63aa7f-bc50-4422-8dc0-e661b521c221-02_505_861_296_602}

1. Write down any pair of mutually exclusive events from \(A , B , C\) and \(D\) Given that \(\mathrm { P } ( B ) = 0.4\)
2. find the value of \(p\) Given also that \(A\) and \(B\) are independent
3. find the value of \(q\) Given further that \(\mathrm { P } \left( B ^ { \prime } \mid C \right) = 0.64\)
4. find
   1. the value of \(r\)
   2. the value of \(s\)

1. A large college produces three magazines.

One magazine is about green issues, one is about equality and one is about sports.
A student at the college is selected at random and the events \(G , E\) and \(S\) are defined as follows \(G\) is the event that the student reads the magazine about green issues \(E\) is the event that the student reads the magazine about equality \(S\) is the event that the student reads the magazine about sports
The Venn diagram, where \(p , q , r\) and \(t\) are probabilities, gives the probability for each subset. \includegraphics[max width=\textwidth, alt={}, center]{10736735-3050-43eb-9e76-011ca6fa48b8-10_508_862_756_603}

1. Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and \(\mathrm { P } ( G ) = 0.25\)
2. Find
   1. the value of \(p\)
   2. the value of \(q\) Given that \(\mathrm { P } ( S \mid E ) = \frac { 5 } { 12 }\)
3. find
   1. the value of \(r\)
   2. the value of \(t\)
4. Determine whether or not the events ( \(S \cap E ^ { \prime }\) ) and \(G\) are independent. Show your working clearly. \section*{Question 4 continued.} \section*{Question 4 continued.} \section*{Question 4 continued.}

12 The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students. \includegraphics[max width=\textwidth, alt={}, center]{68f1107f-f188-4698-934e-8fd593b25418-7_554_910_347_244} A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and the second student studies Geography but not Psychology. \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA

5 Each day John either cycles to work or goes on the bus.

• If it is raining when John is ready to set off for work, the probability that he cycles to work is 0.4.
• If it is not raining when John is ready to set off for work, the probability that he cycles to work is 0.9 .
• The probability that it is raining when he is ready to set off for work is 0.2 .

You should assume that days on which it rains occur randomly and independently.

1. Draw a tree diagram to show the possible outcomes and their associated probabilities.
2. Calculate the probability that, on a randomly chosen day, John cycles to work. John works 5 days each week.
3. Calculate the probability that he cycles to work every day in a randomly chosen working week.

4 There are four human blood groups; these are called \(\mathrm { O } , \mathrm { A } , \mathrm { B }\) and AB . Each person has one of these blood groups. The table below shows the distribution of blood groups in a large country. Two people are selected at random from this country.

1. Find the probability that at least one of these two people has blood group O .
2. Find the probability that each of these two people has a different blood group.