2.03b Probability diagrams: tree, Venn, sample space

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.

5. A biased tetrahedral die has faces numbered \(0,1,2\) and 3 . The die is rolled and the number face down on the die, \(X\), is recorded. The probability distribution of \(X\) is If \(X = 3\) then the final score is 3
If \(X \neq 3\) then the die is rolled again and the final score is the sum of the two numbers. The random variable \(T\) is the final score.

1. Find \(\mathrm { P } ( T = 2 )\)
2. Find \(\mathrm { P } ( T = 3 )\)
3. Given that the die is rolled twice, find the probability that the final score is 3

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.

There are 7 red counters, 3 blue counters and 2 yellow counters in a bag. Gina selects a counter at random from the bag and keeps it. If the counter is yellow she does not select any more counters. If the counter is not yellow she randomly selects a second counter from the bag.

1. Complete the tree diagram. First Counter
   Second Counter \includegraphics[max width=\textwidth, alt={}, center]{a439724e-b570-434d-bf75-de2b50915042-02_1147_1081_603_397} Given that Gina has selected a yellow counter,
2. find the probability that she has 2 counters.
   

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.

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

2. An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability \(\frac { 2 } { 3 }\) of landing heads is spun.
When a blue ball is selected a fair coin is spun.

1. Complete the tree diagram below to show the possible outcomes and associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_785_385_744_568} \includegraphics[max width=\textwidth, alt={}, center]{61983561-79f7-4883-8ae7-ab1f4955d444-04_1054_483_760_954} Shivani selects a ball and spins the appropriate coin.
2. Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
3. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
4. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.

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.
   

4. The employees of a company are classified as management, administration or production. The following table shows the number employed in each category and whether or not they live close to the company or some distance away. An employee is chosen at random.
Find the probability that this employee

1. is an administrator,
2. lives close to the company, given that the employee is a manager. Of the managers, \(90 \%\) are married, as are \(60 \%\) of the administrators and \(80 \%\) of the production employees.
3. Construct a tree diagram containing all the probabilities.
4. Find the probability that an employee chosen at random is married. An employee is selected at random and found to be married.
5. Find the probability that this employee is in production.

2. A car dealer offers purchasers a three year warranty on a new car. He sells two models, the Zippy and the Nifty. For the first 50 cars sold of each model the number of claims under the warranty is shown in the table below. One of the purchasers is chosen at random. Let \(A\) be the event that no claim is made by the purchaser under the warranty and \(B\) the event that the car purchased is a Nifty.

1. Find \(\mathrm { P } ( A \cap B )\).
2. Find \(\mathrm { P } \left( A ^ { \prime } \right)\). Given that the purchaser chosen does not make a claim under the warranty,
3. find the probability that the car purchased is a Zippy.
4. Show that making a claim is not independent of the make of the car purchased. Comment on this result.

A company assembles drills using components from two sources. Goodbuy supplies \(85 \%\) of the components and Amart supplies the rest. It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.

1. Represent this information on a tree diagram. An assembled drill is selected at random.
2. Find the probability that it is not faulty.
   

5. Articles made on a lathe are subject to three kinds of defect, \(A , B\) or \(C\). A sample of 1000 articles was inspected and the following results were obtained. \begin{displayquote} 31 had a type \(A\) defect
37 had a type \(B\) defect
42 had a type \(C\) defect
11 had both type \(A\) and type \(B\) defects
13 had both type \(B\) and type \(C\) defects
10 had both type \(A\) and type \(C\) defects
6 had all three types of defect.

1. Draw a Venn diagram to represent these data. \end{displayquote} Find the probability that a randomly selected article from this sample had
2. no defects,
3. no more than one of these defects. An article selected at random from this sample had only one defect.
4. Find the probability that it was a type \(B\) defect. Two different articles were selected at random from this sample.
5. Find the probability that both had type \(B\) defects.

4. A bag contains 9 blue balls and 3 red balls. A ball is selected at random from the bag and its colour is recorded. The ball is not replaced. A second ball is selected at random and its colour is recorded.

1. Draw a tree diagram to represent the information. Find the probability that
   1. the second ball selected is red,
   2. both balls selected are red, given that the second ball selected is red.

6. For the events \(A\) and \(B\), $$\mathrm { P } \left( A \cap B ^ { \prime } \right) = 0.32 , \mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.11 \text { and } \mathrm { P } ( A \cup B ) = 0.65$$

1. Draw a Venn diagram to illustrate the complete sample space for the events \(A\) and \(B\).
2. Write down the value of \(\mathrm { P } ( A )\) and the value of \(\mathrm { P } ( B )\).
3. Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
4. Determine whether or not \(A\) and \(B\) are independent.