2.03a Mutually exclusive and independent events

7 The table shows the numbers of members of a swimming club in certain categories. It is given that \(\frac { 5 } { 8 }\) of the female members are children.

1. Find the value of \(n\).
2. Find the probability that a member chosen at random is either female or a child (or both). The table below shows the corresponding numbers for an athletics club.

   \cline { 2 - 3 } \multicolumn{1}{c|}{}	Male	Female
   Adults	6	4
   Children	5	10

3. Two members of the athletics club are chosen at random for a photograph.
   1. Find the probability that one of these members is a female child and the other is an adult male.
   2. Find the probability that exactly one of these members is female and exactly one is a child.

4 In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random.

• \(T\) is the event that this person likes tomato soup.
• \(M\) is the event that this person likes mushroom soup.

You are given that \(\mathrm { P } ( T ) = 0.55 , \mathrm { P } ( M ) = 0.33\) and \(\mathrm { P } ( T \mid M ) = 0.80\).

1. Use this information to show that the events \(T\) and \(M\) are not independent.
2. Find \(\mathrm { P } ( T \cap M )\).
3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram.

3 Each weekday Alan drives to work. On his journey, he goes over a level crossing. Sometimes he has to wait at the level crossing for a train to pass.

• \(W\) is the event that Alan has to wait at the level crossing.
• \(L\) is the event that Alan is late for work.

You are given that \(\mathrm { P } ( L \mid W ) = 0.4 , \mathrm { P } ( W ) = 0.07\) and \(\mathrm { P } ( L \cup W ) = 0.08\).

1. Calculate \(\mathrm { P } ( L \cap W )\).
2. Draw a Venn diagram, showing the events \(L\) and \(W\). Fill in the probability corresponding to each of the four regions of your diagram.
3. Determine whether the events \(L\) and \(W\) are independent, explaining your method clearly.

4 Each packet of Cruncho cereal contains one free fridge magnet. There are five different types of fridge magnet to collect. They are distributed, with equal probability, randomly and independently in the packets. Keith is about to start collecting these fridge magnets.

1. Find the probability that the first 2 packets that Keith buys contain the same type of fridge magnet.
2. Find the probability that Keith collects all five types of fridge magnet by buying just 5 packets.
3. Hence find the probability that Keith has to buy more than 5 packets to acquire a complete set.

3 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.

1. Find the probability that neither card is an ace.
2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.

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 \(\mathrm { P } ( C )\).

5 The independent discrete random variables \(U\) and \(V\) can each take the values 1, 2 and 3, all with probability \(\frac { 1 } { 3 }\). The random variables \(X\) and \(Y\) are defined as follows: $$X = | U - V | , Y = U + V .$$

1. In the Printed Answer Book complete the table showing the joint probability distribution of \(X\) and \(Y\).
2. Find \(\operatorname { Cov } ( X , Y )\).
3. State with a reason whether \(X\) and \(Y\) are independent.
4. Find \(\mathrm { P } ( Y = 3 \mid X = 1 )\).

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.

3 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{1ad9c390-b42f-47d8-86c5-f73a42d97721-02_513_826_1713_658} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).

4 Joe is to catch a plane to go on holiday. He has arranged to leave his car at a car park near to the airport. There is a bus service from the car park to the airport, and the bus leaves when there are at least 15 passengers on board. Joe is delayed getting to the car park and arrives needing the bus to leave within 15 minutes if he is to catch his plane. He is the \(10 ^ { \text {th } }\) passenger to board the bus, so he has to wait for another 5 passengers to arrive. The distribution of the time intervals between car arrivals and the distribution of the number of passengers per car are given below.

1. Give an efficient rule for using 2-digit random numbers to simulate the intervals between car arrivals.
2. Give an efficient rule for using 2-digit random numbers to simulate the number of passengers in a car.
3. The incomplete table in your answer book shows the results of nine simulations of the situation. Complete the table, showing in each case whether or not Joe catches his plane.
4. Use the random numbers provided in your answer book to run a tenth simulation.
   [0pt]
5. Estimate the probability of Joe catching his plane. State how you could improve your estimate. [2]

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.

11 Each of the 30 students in a class plays at least one of squash, hockey and tennis.

• 18 students play squash
• 19 students play hockey
• 17 students play tennis
• 8 students play squash and hockey
• 9 students play hockey and tennis
• 11 students play squash and tennis
  1. Find the number of students who play all three sports.

A student is picked at random from the class.

Given that this student plays squash, find the probability that this student does not play hockey. Two different students are picked at random from the class, one after the other, without replacement.

Given that the first student plays squash, find the probability that the second student plays hockey.

1. The Venn diagram shows three events, \(A\), \(B\) and \(C\), and their associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{e73193ee-339e-48ab-811c-9ab6817f786d-04_680_780_296_644}

Events \(B\) and \(C\) are mutually exclusive.
Events \(A\) and \(C\) are independent.
Showing your working, find the value of \(x\), the value of \(y\) and the value of \(z\).

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. 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 \(s\)