2.03c Conditional probability: using diagrams/tables

8 The box and whisker plot below summarises the weights in grams of the 20 chocolates in a box. \includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-4_287_1177_319_427}

1. Find the interquartile range of the data and hence determine whether there are any outliers at either end of the distribution. Ben buys a box of these chocolates each weekend. The chocolates all look the same on the outside, but 7 of them have orange centres, 6 have cherry centres, 4 have coffee centres and 3 have lemon centres. One weekend, each of Ben's 3 children eats one of the chocolates, chosen at random.
2. Calculate the probabilities of the following events. A: all 3 chocolates have orange centres \(B\) : all 3 chocolates have the same centres
3. Find \(\mathrm { P } ( A \mid B )\) and \(\mathrm { P } ( B \mid A )\). The following weekend, Ben buys an identical box of chocolates and again each of his 3 children eats one of the chocolates, chosen at random.
4. Find the probability that, on both weekends, the 3 chocolates that they eat all have orange centres.
5. Ben likes all of the chocolates except those with cherry centres. On another weekend he is the first of his family to eat some of the chocolates. Find the probability that he has to select more than 2 chocolates before he finds one that he likes. \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

7 A particular disease occurs in a proportion \(p\) of the population of a town. A diagnostic test has been developed, in which a positive result indicates the presence of the disease. It has a probability 0.98 of giving a true positive result, i.e. of indicating the presence of the disease when it is actually present. The test will give a false positive result with probability 0.08 when the disease is not present. A randomly chosen person is given the test.

1. Find, in terms of \(p\), the probability that
   1. the person has the disease when the result is positive,
   2. the test will lead to a wrong conclusion. It is decided that if the result of the test on someone is positive, that person is tested again. The result of the second test is independent of the result of the first test.
   3. Find the probability that the person has the disease when the result of the second test is positive.
   4. The town has 24000 children and plans to test all of them at a cost of \(\pounds 5\) per test. Assuming that \(p = 0.001\), calculate the expected total cost of carrying out these tests.

8 For the events \(L\) and \(M , \mathrm { P } ( L \mid M ) = 0.2 , \mathrm { P } ( M \mid L ) = 0.4\) and \(\mathrm { P } ( M ) = 0.6\).

1. Find \(\mathrm { P } ( L )\) and \(\mathrm { P } \left( L ^ { \prime } \cup M ^ { \prime } \right)\).
2. Given that, for the event \(N , \mathrm { P } ( N \mid ( L \cap M ) ) = 0.3\), find \(\mathrm { P } \left( L ^ { \prime } \cup M ^ { \prime } \cup N ^ { \prime } \right)\).

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.

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

6 In winter in Metland the weather each day can be classified as dry, wet or snowy. The table shows the probabilities for the next day's weather given the current day's weather. You are to use two-digit random numbers to simulate the winter weather in Metland.

1. Give an efficient rule for using two-digit random numbers to simulate tomorrow's weather if today is
   (A) dry,
   (B) wet,
   (C) snowy.
2. Today is a dry winter's day in Metland. Use the following two-digit random numbers to simulate the next 7 days' weather in Metland. $$\begin{array} { l l l l l l l l l l } 23 & 85 & 98 & 99 & 56 & 47 & 82 & 14 & 03 & 12 \end{array}$$
3. Use your simulation from part (ii) to estimate the proportion of dry days in a Metland winter.
4. Explain how you could use simulation to produce an improved estimate of the proportion of dry days in a Metland winter.
5. Give two criticisms of this model of weather.

4 The diagram represents a very simple maze with two vertices, A and B. At each vertex a rat either exits the maze or runs to the other vertex, each with probability 0.5 . The rat starts at vertex A . \includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_79_930_534_571}

1. Describe how to use 1-digit random numbers to simulate this situation.
2. Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex, and calculate the mean number of times it runs between vertices before exiting. The second diagram represents a maze with three vertices, A, B and C. At each vertex there are three possibilities, and the rat chooses one, each with probability \(1 / 3\). The rat starts at vertex A. \includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_566_889_1082_589}
3. Describe how to use 1-digit random numbers to simulate this situation.
4. Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex.

5 The diagram shows the progress of a drunkard towards his home on one particular night. For every step which he takes towards his home, he staggers one step diagonally to his left or one step diagonally to his right, randomly and with equal probability. There is a canal three steps to the right of his starting point, and no constraint to the left. On this particular occasion he falls into the canal after 5 steps. \includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-5_723_622_488_724}

1. Explain how you would simulate the drunkard's walk, making efficient use of one-digit random numbers.
2. Using the random digits in the Printed Answer Book simulate the drunkard's walk and show his progress on the grid. Stop your simulation either when he falls into the canal or when he has staggered 6 steps, whichever happens first.
3. How could you estimate the probability of him falling into the canal within 6 steps? On another occasion the drunkard sets off carrying a briefcase in his right hand. This changes the probabilities of him staggering to the right to \(\frac { 2 } { 3 }\), and to the left to \(\frac { 1 } { 3 }\).
4. Explain how you would now simulate this situation.
5. Simulate the drunkard's walk (with briefcase) 10 times, and hence estimate the probability of him falling into the canal within 6 steps. (In your simulations you are not required to show his progress on a grid. You only need to record his steps to the right or left.)

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.

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. Find

$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - 6 \sqrt { x } + 1 \right) \mathrm { d } x$$ giving your answer in its simplest form.

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 \(z\).

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. Two bags, \(\mathbf { A }\) and \(\mathbf { B }\), each contain balls which are either red or yellow or green.

Bag A contains 4 red, 3 yellow and \(n\) green balls.
Bag \(\mathbf { B }\) contains 5 red, 3 yellow and 1 green ball.
A ball is selected at random from bag \(\mathbf { A }\) and placed into bag \(\mathbf { B }\).
A ball is then selected at random from bag \(\mathbf { B }\) and placed into bag \(\mathbf { A }\).
The probability that bag \(\mathbf { A }\) now contains an equal number of red, yellow and green balls is \(p\). Given that \(p > 0\), find the possible values of \(n\) and \(p\).

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. Tisam is playing a game.

She uses a ball, a cup and a spinner.
The random variable \(X\) represents the number the spinner lands on when it is spun. The probability distribution of \(X\) is given in the following table where \(a , b , c\) and \(d\) are probabilities.
To play the game

• the spinner is spun to obtain a value of \(x\)
• Tisam then stands \(x \mathrm {~cm}\) from the cup and tries to throw the ball into the cup

The event \(S\) represents the event that Tisam successfully throws the ball into the cup.
To model this game Tisam assumes that

• \(\mathrm { P } ( S \mid \{ X = x \} ) = \frac { k } { x }\) where \(k\) is a constant
• \(\mathrm { P } ( S \cap \{ X = x \} )\) should be the same whatever value of \(x\) is obtained from the spinner

Using Tisam's model,

1. show that \(c = \frac { 8 } { 5 } b\)
2. find the probability distribution of \(X\) Nav tries, a large number of times, to throw the ball into the cup from a distance of 100 cm .
   He successfully gets the ball in the cup \(30 \%\) of the time.
3. State, giving a reason, why Tisam's model of this game is not suitable to describe Nav playing the game for all values of \(X\)

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.}

10 Jane conducted a survey. She chose a sample of people from three towns, A, B and C. She noted the following information. 400 people were chosen.
230 people were adults.
55 adults were from town A .
65 children were from town A .
35 children were from town B .
150 people were from town B .

1. In the Printed Answer Booklet, complete the two-way frequency table.

   \multirow{2}{*}{}	Town
   	A	B	C	Total
   adult
   child
   Total

2. One of the people is chosen at random.
   1. Find the probability that this person is an adult from town A .
   2. Given that the person is from town A , find the probability that the person is an adult. For another survey, Jane wanted to choose a random sample from the 820 students living in a particular hostel. She numbered the students from 1 to 820 and then generated some random numbers on her calculator. The random numbers were 0.114287562 and 0.081859817 . Jane's friend Kareem used these figures to write down the following sample of five student numbers. 114, 142, 428, 287 and 756 Jane used the same figures to write down the following sample of five student numbers.
      114, 287, 562, 81 and 817
3. 1. State, with a reason, which one of these samples is not random.
   2. Explain why Jane omitted the number 859 from her sample.

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.

8 A team called "The Educated Guess" enter a weekly quiz. If they win the quiz in a particular week, the probability that they will win the following week is 0.4 , but if they do not win, the probability that they will win the following week is 0.2 . In week 4 The Educated Guess won the quiz.

1. Calculate the probability that The Educated Guess will win the quiz in week 6. Every week the same 20 quiz teams, each with 6 members, take part in a quiz. Every member of every team buys a raffle ticket. Five winning tickets are drawn randomly, without replacement. Alf, who is a member of one of the teams, takes part every week.
2. Calculate the probability that, in a randomly chosen week, Alf wins a raffle prize.
3. Find the smallest number of weeks after which it will be \(95 \%\) certain that Alf has won at least one raffle prize.

16 Research conducted by social scientists has shown that \(16 \%\) of young adults smoke cigarettes. Two young adults are selected at random.

1. Determine the probability that one smokes cigarettes and the other doesn't. The same research has also shown that

7 You are given that \(P ( A ) = 0.6 , P ( B ) = 0.5\) and \(P ( A \cup B ) ^ { \prime } = 0.2\).

1. Find \(\mathrm { P } ( \mathrm { A } \cap \mathrm { B } )\).
2. Find \(\mathrm { P } ( \mathrm { A } \mid \mathrm { B } )\).
3. State, with a reason, whether \(A\) and \(B\) are independent.