2.03b Probability diagrams: tree, Venn, sample space

2 Each day the probability that Ashwin wears a tie is 0.2 . The probability that he wears a jacket is 0.4 . If he wears a jacket, the probability that he wears a tie is 0.3 .

1. Find the probability that, on a randomly selected day, Ashwin wears a jacket and a tie.
2. Draw a Venn diagram, using one circle for the event 'wears a jacket' and one circle for the event 'wears a tie'. Your diagram should include the probability for each region.
3. Using your Venn diagram, or otherwise, find the probability that, on a randomly selected day, Ashwin
   (A) wears either a jacket or a tie (or both),
   (B) wears no tie or no jacket (or wears neither).

4 It has been estimated that \(90 \%\) of paintings offered for sale at a particular auction house are genuine, and that the other \(10 \%\) are fakes. The auction house has a test to determine whether or not a given painting is genuine. If this test gives a positive result, it suggests that the painting is genuine. A negative result suggests that the painting is a fake. If a painting is genuine, the probability that the test result is positive is 0.95 .
If a painting is a fake, the probability that the test result is positive is 0.2 .

1. Copy and complete the probability tree diagram below, to illustrate the information above. \includegraphics[max width=\textwidth, alt={}, center]{f3d936ba-8f60-4350-a5b3-92200996434c-3_466_667_834_737} Calculate the probabilities of the following events.
2. The test gives a positive result.
3. The test gives a correct result.
4. The painting is genuine, given a positive result.
5. The painting is a fake, given a negative result. A second test is more accurate, but very expensive. The auction house has a policy of only using this second test on those paintings with a negative result on the original test.
6. Using your answers to parts (iv) and (v), explain why the auction house has this policy. The probability that the second test gives a correct result is 0.96 whether the painting is genuine or a fake.
7. Three paintings are independently offered for sale at the auction house. Calculate the probability that all three paintings are genuine, are judged to be fakes in the first test, but are judged to be genuine in the second test.

5. A group of 100 students are asked if they like folk music, rock music or soul music. \begin{displayquote} All students who like folk music also like rock music No students like both rock music and soul music 75 students do not like soul music 12 students who like rock music do not like folk music 30 students like folk music

1. Draw a Venn diagram to illustrate this information.
2. State two of these types of music that are mutually exclusive. \end{displayquote} Find the probability that a randomly chosen student
3. does not like folk music, rock music or soul music,
4. likes rock music,
5. likes folk music or soul music. Given that a randomly chosen student likes rock music,
6. find the probability that he or she also likes folk music.

4. Events \(A\) and \(B\) are shown in the Venn diagram below
where \(x , y , 0.10\) and 0.32 are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{c58f3e88-2dbc-40d6-a966-a5765a7c67ba-08_467_798_408_575}

1. Find an expression in terms of \(x\) for
   1. \(\mathrm { P } ( A )\)
   2. \(\mathrm { P } ( B \mid A )\)
2. Find an expression in terms of \(x\) and \(y\) for \(\mathrm { P } ( A \cup B )\) Given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\)
3. find the value of \(x\) and the value of \(y\)

4. A training agency awards a certificate to each student who passes a test while completing a course.
Students failing the test will attempt the test again up to 3 more times, and, if they pass the test, will be awarded a certificate.
The probability of passing the test at the first attempt is 0.7 , but the probability of passing reduces by 0.2 at each attempt.

1. Complete the tree diagram below to show this information. \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-08_545_1244_639_340} A student who completed the course is selected at random.
2. Find the probability that the student was awarded a certificate.
3. Given that the student was awarded a certificate, find the probability that the student passed on the first or second attempt. The training agency decides to alter the test taken by the students while completing the course, but will not allow more than 2 attempts. The agency requires the probability of passing the test at the first attempt to be \(p\), and the probability of passing the test at the second attempt to be ( \(p - 0.2\) ). The percentage of students who complete the course and are awarded a certificate is to be \(95 \%\)
4. Show that \(p\) satisfies the equation $$p ^ { 2 } - 2.2 p + 1.15 = 0$$
5. Hence find the value of \(p\), giving your answer to 3 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-09_2261_47_313_37}

1. In a survey, people were asked if they use a computer every day.

Of those people under 50 years old, \(80 \%\) said they use computer every day. Of those people aged 50 or more, \(55 \%\) said they use computer every day. The proportion of people in the survey under 50 years old is \(p\)

1. Draw a tree diagram to represent this information. In the survey, 70\% of all people said they use computer every day.
2. Find the value of \(p\) One person is selected at random. Given that this person uses a computer every day,
3. find the probability that this person is under 50 years old.
   \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

1. Anju has a bag that contains 5 socks of which 2 are blue.

Anju randomly selects socks from the bag, one sock at a time. She does not replace any socks but continues to select socks at random until she has both blue socks. The discrete random variable \(S\) represents the total number of socks that Anju has selected.

1. Write down the value of \(\mathrm { P } ( S = 1 )\)
2. Find \(\mathrm { P } ( S > 2 )\)
3. Find \(\mathrm { P } ( S = 3 )\)
4. Given that the second sock selected is blue, find the probability that Anju selects exactly 3 socks.
5. Find \(\mathrm { P } ( S = 5 )\)

1. The Venn diagram shows the events \(A , B\) and \(C\) and their associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{4f034b9a-94c8-42f2-bd77-9adec277aba6-02_584_1061_296_445}

Find

1. \(\mathrm { P } \left( B ^ { \prime } \right)\)
2. \(\mathrm { P } ( A \cup C )\)
3. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\)

1. Two bags, \(\boldsymbol { X }\) and \(\boldsymbol { Y }\), each contain green marbles (G) and blue marbles (B) only.

• Bag \(\boldsymbol { X }\) contains 5 green marbles and 4 blue marbles
• Bag \(\boldsymbol { Y }\) contains 6 green marbles and 5 blue marbles

A marble is selected at random from bag \(\boldsymbol { X }\) and placed in bag \(\boldsymbol { Y }\) A second marble is selected at random from bag \(\boldsymbol { X }\) and placed in bag \(\boldsymbol { Y }\) A third marble is then selected, this time from bag \(\boldsymbol { Y }\)

1. Use this information to complete the tree diagram shown on page 7
2. Find the probability that the 2 marbles selected from bag \(\boldsymbol { X }\) are of different colours.
3. Find the probability that all 3 marbles selected are the same colour. Given that all three marbles selected are the same colour,
4. find the probability that they are all green. 2nd Marble (from bag \(\boldsymbol { X }\) ) \section*{3rd Marble (from bag Y)} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{1st Marble (from bag \(\boldsymbol { X }\) )} \includegraphics[alt={},max width=\textwidth]{c316fa29-dedc-4890-bd82-31eb0bb819f9-07_1694_1312_484_310}
   \end{figure}

1. 1. In the Venn diagram below, \(A\) and \(B\) represent events and \(p , q , r\) and \(s\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-12_400_789_347_639}

$$\mathrm { P } ( A ) = \frac { 7 } { 25 } \quad \mathrm { P } ( B ) = \frac { 1 } { 5 } \quad \mathrm { P } \left[ \left( A \cap B ^ { \prime } \right) \cup \left( A ^ { \prime } \cap B \right) \right] = \frac { 8 } { 25 }$$

1. Use algebra to show that \(2 p + 2 q + 2 r = \frac { 4 } { 5 }\)
2. Find the value of \(p\), the value of \(q\), the value of \(r\) and the value of \(s\) (ii) Two events, \(C\) and \(D\), are such that $$\mathrm { P } ( C ) = \frac { x } { x + 5 } \quad \mathrm { P } ( D ) = \frac { 5 } { x }$$ where \(x\) is a positive constant.
   By considering \(\mathrm { P } ( C ) + \mathrm { P } ( D )\) show that \(C\) and \(D\) cannot be mutually exclusive.

1. In a sixth form college each student in Year 12 and Year 13 is either left-handed (L) or right-handed (R).

The partially completed tree diagram, where \(p\) is a probability, gives information about these students. \includegraphics[max width=\textwidth, alt={}, center]{86446ce3-496a-4f02-9566-9b207bac9efa-10_960_981_477_543}

1. Complete the tree diagram, in terms of \(p\) where necessary. The probability that a student is left-handed is 0.11
2. Find the value of \(p\)
3. Find the probability that a student selected at random is in Year 12 and left-handed. Given that a student is right-handed,
4. find the probability that the student is in Year 12

6. The Venn diagram below shows the probabilities of customers having various combinations of a starter, main course or dessert at Polly's restaurant. \(S =\) the event a customer has a starter. \(M =\) the event a customer has a main course. \(D =\) the event a customer has a dessert. \includegraphics[max width=\textwidth, alt={}, center]{fa0dbe16-ace8-4c44-8404-2bc4e1879d57-10_602_1125_607_413} Given that the events \(S\) and \(D\) are statistically independent

1. find the value of \(p\).
2. Hence find the value of \(q\).
3. Find
   1. \(\quad\) P( \(D \mid M \cap S\) )
   2. \(\operatorname { P } \left( D \mid M \cap S ^ { \prime } \right)\) One evening 63 customers are booked into Polly's restaurant for an office party. Polly has asked for their starter and main course orders before they arrive. Of these 63 customers 27 ordered a main course and a starter, 36 ordered a main course without a starter.
4. Estimate the number of desserts that these 63 customers will have.

4. \(\quad\) The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 2 } { 5 } , \mathrm { P } ( B ) = \frac { 1 } { 2 }\) and \(\mathrm { P } \left( A \quad B ^ { \prime } \right) = \frac { 4 } { 5 }\).

1. Find
   1. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
   2. \(\mathrm { P } ( A \cap B )\),
   3. \(\mathrm { P } ( A \cup B )\),
   4. \(\mathrm { P } \left( \begin{array} { l l } A & B \end{array} \right)\).
2. State, with a reason, whether or \(\operatorname { not } A\) and \(B\) are
   1. mutually exclusive,
   2. independent.

6. One of the objectives of a computer game is to collect keys. There are three stages to the game. The probability of collecting a key at the first stage is \(\frac { 2 } { 3 }\), at the second stage is \(\frac { 1 } { 2 }\), and at the third stage is \(\frac { 1 } { 4 }\).

1. Draw a tree diagram to represent the 3 stages of the game.
2. Find the probability of collecting all 3 keys.
3. Find the probability of collecting exactly one key in a game.
4. Calculate the probability that keys are not collected on at least 2 successive stages in a game.

7 Bag \(A\) contains 4 balls numbered 2, 4, 5, 8. Bag \(B\) contains 5 balls numbered 1, 3, 6, 8, 8. Bag \(C\) contains 7 balls numbered \(2,7,8,8,8,8,9\). One ball is selected at random from each bag.

• Event \(X\) is 'exactly two of the selected balls have the same number'.
• Event \(Y\) is 'the ball selected from bag \(A\) has number 4'.
  1. Find \(\mathrm { P } ( X )\).
  2. Find \(\mathrm { P } ( X \cap Y )\) and hence determine whether or not events \(X\) and \(Y\) are independent.
  3. Find the probability that two balls are numbered 2, given that exactly two of the selected balls have the same number.
     

5 A bag contains 4 blue discs and 6 red discs. Chloe takes a disc from the bag. If this disc is red, she takes 2 more discs. If not, she takes 1 more disc. Each disc is taken at random and no discs are replaced.

1. Complete the probability tree diagram in your Answer Book, showing all the probabilities. \includegraphics[max width=\textwidth, alt={}, center]{48ffcd44-d933-40e0-818a-20d6db607298-4_730_1203_529_511} The total number of blue discs that Chloe takes is denoted by \(X\).
2. Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 5 }\). The complete probability distribution of \(X\) is given below.

   \(x\)	0	1	2
   \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 6 }\)	\(\frac { 3 } { 5 }\)	\(\frac { 7 } { 30 }\)

3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

8 Ann, Bill, Chris and Dipak play a game with a fair cubical die. Starting with Ann they take turns, in alphabetical order, to throw the die. This process is repeated as many times as necessary until a player throws a 6 . When this happens, the game stops and this player is the winner. Find the probability that

1. Chris wins on his first throw,
2. Dipak wins on his second throw,
3. Ann gets a third throw,
4. Bill throws the die exactly three times.

4 A bag contains 5 red discs and 1 black disc. Tina takes two discs from the bag at random without replacement.

1. The diagram shows part of a tree diagram to illustrate this situation. \section*{First disc}
   \includegraphics[max width=\textwidth, alt={}]{e23cb28b-49e5-436a-942d-e6320029c634-3_264_494_479_550}
   Complete the tree diagram in your Answer Book showing all the probabilities. \section*{Second disc}
2. Find the probability that exactly one of the two discs is red. All the discs are replaced in the bag. Tony now takes three discs from the bag at random without replacement.
3. Given that the first disc Tony takes is red, find the probability that the third disc Tony takes is also red.

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.

8 A game is played with a fair, six-sided die which has 4 red faces and 2 blue faces. One turn consists of throwing the die repeatedly until a blue face is on top or until the die has been thrown 4 times.

1. In the answer book, complete the probability tree diagram for one turn. \includegraphics[max width=\textwidth, alt={}, center]{e5957185-5fe3-45d9-9ab3-c2aab9cbd8dd-5_314_302_1000_884}
2. Find the probability that in one particular turn the die is thrown 4 times.
3. Adnan and Beryl each have one turn. Find the probability that Adnan throws the die more times than Beryl.
4. State one change that needs to be made to the rules so that the number of throws in one turn will have a geometric distribution.

3 Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is

• 0.7 if he won the previous set,
• 0.4 if Alan won the previous set.

It is not possible to draw a set.

1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets.
2. Find the probability that Alan wins the match.
3. Find the probability that the match ends after exactly four sets have been played.

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.

6 Whitefly, blight and mosaic virus are three problems which can affect tomato plants. 100 tomato plants are examined for these problems. The numbers of plants with each type of problem are shown in the Venn diagram. 47 of the plants have none of the problems. \includegraphics[max width=\textwidth, alt={}, center]{3a5d18f5-b1fe-4513-ae4e-f37c20f172b5-3_668_812_998_664}

1. One of the 100 plants is selected at random. Find the probability that this plant has
   (A) at most one of the problems,
   (B) exactly two of the problems.
2. Three of the 100 plants are selected at random. Find the probability that all of them have at least one of the problems. Section B (36 marks)

7 Laura frequently flies to business meetings and often finds that her flights are delayed. A flight may be delayed due to technical problems, weather problems or congestion problems, with probabilities \(0.2,0.15\) and 0.1 respectively. The tree diagram shows this information. \includegraphics[max width=\textwidth, alt={}, center]{3a5d18f5-b1fe-4513-ae4e-f37c20f172b5-4_608_1651_532_248}

1. Write down the values of the probabilities \(a , b\) and \(c\) shown in the tree diagram. One of Laura's flights is selected at random.
2. Find the probability that Laura's flight is not delayed and hence write down the probability that it is delayed.
3. Find the probability that Laura's flight is delayed due to just one of the three problems.
4. Given that Laura's flight is delayed, find the probability that the delay is due to just one of the three problems.
5. Given that Laura's flight has no technical problems, find the probability that it is delayed.
6. In a particular year, Laura has 110 flights. Find the expected number of flights that are delayed.