2.03a Mutually exclusive and independent events

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

13

1. The probability distribution of a random variable \(X\) is shown in the table, where \(p\) is a constant.

   \(x\)	0	1	2	3
   \(P ( X = x )\)	\(\frac { 1 } { 12 }\)	\(\frac { 1 } { 4 }\)	\(p\)	\(3 p\)

   Two values of \(X\) are chosen at random. Determine the probability that their product is greater than their sum.
2. A random variable \(Y\) takes \(n\) values, each of which is equally likely. Two values, \(Y _ { 1 }\) and \(Y _ { 2 }\), of \(Y\) are chosen at random. It is given that \(\mathrm { P } \left( Y _ { 1 } = Y _ { 2 } \right) = 0.02\).
   Find \(\mathrm { P } \left( Y _ { 1 } > Y _ { 2 } \right)\).

9 A fair six-sided die has its faces numbered 1, 3, 4, 5, 6 and 7. The die is rolled once. \(A\) is the event that the die shows an even number. \(B\) is the event that the die shows a prime number.

1. Write down the value of \(\mathrm { p } ( A )\).
2. Write down the value of \(\mathrm { p } ( B )\).
3. Write down the value of \(\mathrm { p } ( A\) or \(B )\). The die is rolled again.
4. Calculate the probability that the sum of the scores from the two rolls is even.

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.

3 The probability that Chipping FC win a league football match is \(\mathrm { P } ( W ) = 0.4\).

1. Calculate the probability that Chipping FC fail to win each of their next two league football matches. The probability that Chipping FC lose a league football match is \(\mathrm { P } ( L ) = 0.3\).
2. Explain why \(\mathrm { P } ( W ) + \mathrm { P } ( L ) \neq 1\).

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.

9 Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table. Four puppies are chosen at random to train as guide dogs.

1. Determine the probability that exactly 3 females are chosen.
2. Determine the probability that at least 3 black puppies are chosen.
3. Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen.
4. Explain whether the 2 events 'choosing exactly 3 females' and 'choosing at least 3 black puppies' are independent events.

6. Three events \(A , B\) and \(C\) are such that $$\mathrm { P } ( A ) = \frac { 2 } { 5 } \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 8 }$$ Given that \(A\) and \(C\) are mutually exclusive find

1. \(\mathrm { P } ( A \cup C )\) Given that \(A\) and \(B\) are independent
2. show that \(\mathrm { P } ( B ) = \frac { 3 } { 8 }\)
3. Find \(\mathrm { P } ( A \mid B )\) Given that \(\mathrm { P } \left( C ^ { \prime } \cap B ^ { \prime } \right) = 0.3\)
4. draw a Venn diagram to represent the events \(A , B\) and \(C\)

1. A manufacturer of electric generators buys engines for its generators from three companies, \(R , S\) and \(T\).

Company \(R\) supplies 40\% of the engines. Company \(S\) supplies \(25 \%\) of the engines. The rest of the engines are supplied by company \(T\). It is known that \(2 \%\) of the engines supplied by company \(R\) are faulty, \(1 \%\) of the engines supplied by company \(S\) are faulty and \(2 \%\) of the engines supplied by company \(T\) are faulty. An engine is chosen at random.

1. Draw a tree diagram to show all the possible outcomes and the associated probabilities.
2. Calculate the probability that the engine is from company \(R\) and is not faulty.
3. Calculate the probability that the engine is faulty. Given that the engine is faulty,
4. find the probability that the engine did not come from company \(S\).
   

1. Events \(A\) and \(B\) are such that

$$\mathrm { P } ( A ) = 0.5 \quad \mathrm { P } ( A \mid B ) = \frac { 2 } { 3 } \quad \mathrm { P } \left( A ^ { \prime } \cup B ^ { \prime } \right) = 0.6$$

1. Find \(\mathrm { P } ( B )\)
2. Find \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\) The event \(C\) has \(\mathrm { P } ( C ) = 0.15\) The events \(A\) and \(C\) are mutually exclusive. The events \(B\) and \(C\) are independent.
3. Find \(\mathrm { P } ( B \cap C )\)
4. Draw a Venn diagram to illustrate the events \(A , B\) and \(C\) and the probabilities for each region.
   

3. A certain disease occurs in a population in 2 mutually exclusive types. It is difficult to diagnose people with type \(A\) of the disease and there is an unknown proportion \(p\) of the population with type \(A\).
It is easier to diagnose people with type \(B\) of the disease and it is known that \(2 \%\) of the population have type \(B\). A test has been developed to help diagnose whether or not a person has the disease. The event \(T\) represents a positive result on the test. After a large-scale trial of the test, the following information was obtained. For a person with type \(B\) of the disease the probability of a positive test result is 0.96 For a person who does not have the disease the probability of a positive test result is 0.05 For a person with type \(A\) of the disease the probability of a positive test result is \(q\)

1. Complete the tree diagram. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-08_776_965_1050_484} The probability of a randomly selected person having a positive test result is 0.169 For a person with a positive test result, the probability that they do not have the disease is \(\frac { 41 } { 169 }\)
2. Find the value of \(p\) and the value of \(q\). A doctor is about to see a person who she knows does not have type \(B\) of the disease but does have a positive test result.
3. 1. Find the probability that this person has type \(A\) of the disease.
   2. State, giving a reason, whether or not the doctor will find the test useful.

1. In a school canteen, students can choose from a main course of meat ( \(M\) ), fish ( \(F\) ) or vegetarian ( \(V\) ). They can then choose a drink of either water ( \(W\) ) or juice ( \(J\) ).

The partially completed tree diagram, where \(p\) and \(q\) are probabilities, shows the probabilities of these choices for a randomly selected student. \section*{Drink} \begin{figure}[h] \end{figure}

1. Complete the tree diagram, giving your answers in terms of \(p\) and \(q\) where appropriate.
2. Find an expression, in terms of \(p\) and \(q\), for the probability that a randomly selected student chooses water to drink. The events "choosing a vegetarian main course" and "choosing water to drink" are independent.
3. Find a linear equation in terms of \(p\) and \(q\). A student who has chosen juice to drink is selected at random. The probability that they chose fish for their main course is \(\frac { 7 } { 30 }\)
4. Find the value of \(p\) and the value of \(q\). The canteen manager claims that students who choose water to drink are most likely to choose a fish main course.
5. State, showing your working clearly, whether or not the manager's claim is correct.

2. In the Venn diagram below, \(A , B\) and \(C\) are events and \(p , q , r\) and \(s\) are probabilities. The events \(A\) and \(C\) are independent and \(\mathrm { P } ( A ) = 0.65\) \includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-04_373_815_397_568}

1. State which two of the events \(A\), \(B\) and \(C\) are mutually exclusive.
2. Find the value of \(r\) and the value of \(s\). The events ( \(A \cap C ^ { \prime }\) ) and ( \(B \cup C\) ) are also independent.
3. Find the exact value of \(p\) and the exact value of \(q\). Give your answers as fractions.

1. The events \(H\) and \(W\) are such that

$$\mathrm { P } ( H ) = \frac { 3 } { 8 } \quad \mathrm { P } ( H \cup W ) = \frac { 3 } { 4 }$$ Given that \(H\) and \(W\) are independent,

1. show that \(\mathrm { P } ( W ) = \frac { 3 } { 5 }\) The event \(N\) is such that $$\mathrm { P } ( N ) = \frac { 1 } { 15 } \quad \mathrm { P } ( H \cap N ) = \mathrm { P } ( N )$$
2. Find \(\mathrm { P } \left( N ^ { \prime } \mid H \right)\) Given that \(W\) and \(N\) are mutually exclusive,
3. draw a Venn diagram to represent the events \(H , W\) and \(N\) giving the exact probabilities of each region in the Venn diagram.

The Venn diagram shows the probabilities related to teenagers playing 3 particular board games. \(C\) is the event that a teenager plays Chess \(S\) is the event that a teenager plays Scrabble \(G\) is the event that a teenager plays Go
where \(p\) and \(q\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-22_684_935_598_566}

1. Find the probability that a randomly selected teenager plays Chess but does not play Go. Given that the events \(C\) and \(S\) are independent,
2. find the value of \(p\)
3. Hence find the value of \(q\)
4. Find (i) \(\mathrm { P } \left( ( C \cup S ) \cap G ^ { \prime } \right)\) (ii) \(\mathrm { P } ( C \mid ( S \cap G ) )\) A youth club consists of a large number of teenagers.
   In this youth club 76 teenagers play Chess and Go.
5. Use the information in the Venn diagram to estimate how many of the teenagers in the youth club do not play Scrabble.

1. Hugo recorded the purchases of 80 customers in the ladies fashion department of a large store. His results were as follows

20 customers bought a coat
12 customers bought a coat and a scarf
23 customers bought a pair of gloves
13 customers bought a pair of gloves and a scarf no customer bought a coat and a pair of gloves 14 customers did not buy a coat nor a scarf nor a pair of gloves.

1. Draw a Venn diagram to represent all of this information.
2. One of the 80 customers is selected at random.
   1. Find the probability that the customer bought a scarf.
   2. Given that the customer bought a coat, find the probability that the customer also bought a scarf.
   3. State, giving a reason, whether or not the event 'the customer bought a coat' and the event 'the customer bought a scarf' are statistically independent. Hugo had asked the member of staff selling coats and the member of staff selling gloves to encourage customers also to buy a scarf.
3. By considering suitable conditional probabilities, determine whether the member of staff selling coats or the member of staff selling gloves has the better performance at selling scarves to their customers. Give a reason for your answer.

3. The parking times, \(t\) hours, for cars in a car park are summarised below. $$\text { (You may use } \sum \mathrm { fm } = 182 \text { and } \sum \mathrm { fm } ^ { 2 } = 883 \text { ) }$$ A histogram is drawn to represent these data.
The bar representing the time \(1 \leqslant t < 2\) has a width of 1.5 cm and a height of 6 cm .

1. Calculate the width and the height of the bar representing the time \(4 \leqslant t < 6\)
2. Use linear interpolation to estimate the median parking time for the cars in the car park.
3. Estimate the mean and the standard deviation of the parking time for the cars in the car park.
4. Describe, giving a reason, the skewness of the data. One of these cars is selected at random.
5. Estimate the probability that this car is parked for more than 75 minutes.

4. Pieces of wood cladding are produced by a timber merchant. There are three types of fault, \(A , B\) and \(C\), that can appear in each piece of wood cladding. The Venn diagram shows the probabilities of a piece of wood cladding having the various types of fault. \includegraphics[max width=\textwidth, alt={}, center]{0377c6e9-ab4f-477d-9236-0732fe81f25e-14_602_1120_497_413} A piece of wood cladding is chosen at random.

1. Find the probability that the piece of wood cladding has more than one type of fault. Fault types \(A\) and \(C\) occur independently.
2. Find the probability that the piece of wood cladding has no faults. Given that the piece of wood cladding has fault \(A\),
3. find the probability that it also has fault \(B\) but not fault \(C\). Two pieces of the wood cladding are selected at random.
4. Find the probability that both have exactly 2 types of fault.

The Venn diagram shows the events \(A , B , C\) and \(D\), where \(p , q , r\) and \(s\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{1fda59cb-059e-4850-810f-cc3e69bc058e-20_504_826_296_621}

1. Write down the value of
   1. \(\mathrm { P } ( A )\)
   2. \(\mathrm { P } ( A \mid B )\)
   3. \(\mathrm { P } ( A \mid C )\) Given that \(\mathrm { P } \left( B ^ { \prime } \cap D ^ { \prime } \right) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid D ) = \frac { 3 } { 5 }\)
2. find the exact value of \(q\) and the exact value of \(r\) Given also that \(\mathrm { P } \left( B \cup C ^ { \prime } \right) = \frac { 5 } { 8 }\)
3. find the exact value of \(s\)

1. Sally plays a game in which she can either win or lose.

A turn consists of up to 3 games. On each turn Sally plays the game up to 3 times. If she wins the first 2 games or loses the first 2 games, then she will not play the 3rd game.

• The probability that Sally wins the first game in a turn is 0.7
• If Sally wins a game the probability that she wins the next game is 0.6
• If Sally loses a game the probability that she wins the next game is 0.2
  1. Use this information to complete the tree diagram on page 3
  2. Find the probability that Sally wins the first 2 games in a turn.
  3. Find the probability that Sally wins exactly 2 games in a turn.

Given that Sally wins 2 games in a turn,

find the probability that she won the first 2 games. Given that Sally won the first game in a turn,

find the probability that she won 2 games. 1st game 2nd game win

6. Three events \(A , B\) and \(C\) are such that $$\mathrm { P } ( A ) = \frac { 2 } { 5 } \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 8 }$$ Given that \(A\) and \(C\) are mutually exclusive find

1. \(\mathrm { P } ( A \cup C )\) Given that \(A\) and \(B\) are independent
2. show that \(\mathrm { P } ( B ) = \frac { 3 } { 8 }\)
3. Find \(\mathrm { P } ( A \mid B )\) Given that \(\mathrm { P } \left( C ^ { \prime } \cap B ^ { \prime } \right) = 0.3\)
4. draw a Venn diagram to represent the events \(A , B\) and \(C\) \includegraphics[max width=\textwidth, alt={}, center]{b7500cc1-caa6-4767-bb2e-e3d70474e805-21_2260_53_312_33} \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
   VJYV SIHI NI JIIIM ION OC
   VJYV SIHI NI JLIYM ION OC

1. The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines \(A , B\) and \(C\).

\begin{figure}[h] \end{figure} One of these students is selected at random.

1. Show that the probability that the student reads more than one magazine is \(\frac { 1 } { 6 }\).
2. Find the probability that the student reads \(A\) or \(B\) (or both).
3. Write down the probability that the student reads both \(A\) and \(C\). Given that the student reads at least one of the magazines,
4. find the probability that the student reads \(C\).
5. Determine whether or not reading magazine \(B\) and reading magazine \(C\) are statistically independent.