2.03c Conditional probability: using diagrams/tables

3 A box contains 3 red balls and 5 blue balls. One ball is taken at random from the box and not replaced. A yellow ball is then put into the box. A second ball is now taken at random from the box.

1. Complete the tree diagram to show all the outcomes and the probability for each branch. First ball
   Second ball \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_655_392_688_474} \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_785_387_703_1110}
2. Find the probability that the two balls taken are the same colour.
3. Find the probability that the first ball taken is red, given that the second ball taken is blue.

1 When Shona goes to college she either catches the bus with probability 0.8 or she cycles with probability 0.2 . If she catches the bus, the probability that she is late is 0.4 . If she cycles, the probability that she is late is \(x\). The probability that Shona is not late for college on a randomly chosen day is 0.63 . Find the value of \(x\).

4 In a probability distribution the random variable \(X\) takes the values \(- 1,0,1,2,4\). The probability distribution table for \(X\) is as follows.

1. Find the value of \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Given that \(X\) is greater than zero, find the probability that \(X\) is equal to 2 .

2 Benju cycles to work each morning and he has two possible routes. He chooses the hilly route with probability 0.4 and the busy route with probability 0.6 . If he chooses the hilly route, the probability that he will be late for work is \(x\) and if he chooses the busy route the probability that he will be late for work is \(2 x\). The probability that Benju is late for work on any day is 0.36 .

1. Show that \(x = 0.225\).
2. Given that Benju is not late for work, find the probability that he chooses the hilly route.

6 A box contains 3 red balls and 5 white balls. One ball is chosen at random from the box and is not returned to the box. A second ball is now chosen at random from the box.

1. Find the probability that both balls chosen are red.
2. Show that the probability that the balls chosen are of different colours is \(\frac { 15 } { 28 }\).
3. Given that the second ball chosen is red, find the probability that the first ball chosen is red.
   The random variable \(X\) denotes the number of red balls chosen.
4. Draw up the probability distribution table for \(X\).
5. Find \(\operatorname { Var } ( X )\).

3 It was found that \(68 \%\) of the passengers on a train used a cell phone during their train journey. Of those using a cell phone, \(70 \%\) were under 30 years old, \(25 \%\) were between 30 and 65 years old and the rest were over 65 years old. Of those not using a cell phone, \(26 \%\) were under 30 years old and \(64 \%\) were over 65 years old.

1. Draw a tree diagram to represent this information, giving all probabilities as decimals.
2. Given that one of the passengers is 45 years old, find the probability of this passenger using a cell phone during the journey.

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.

1. Find the probability that exactly two of the selected balls have the same number.
2. Given that exactly two of the selected balls have the same number, find the probability that they are both numbered 2 .
3. Event \(X\) is 'exactly two of the selected balls have the same number'. Event \(Y\) is 'the ball selected from bag \(A\) has number 2'. Showing your working, determine whether events \(X\) and \(Y\) are independent or not.

2 In a group of 30 teenagers, 13 of the 18 males watch 'Kops are Kids' on television and 3 of the 12 females watch 'Kops are Kids'.

1. Find the probability that a person chosen at random from the group is either female or watches 'Kops are Kids' or both.
2. Showing your working, determine whether the events 'the person chosen is male' and 'the person chosen watches Kops are Kids' are independent or not.

6 Ana meets her friends once every day. For each day the probability that she is early is 0.05 and the probability that she is late is 0.75 . Otherwise she is on time.

1. Find the probability that she is on time on fewer than 20 of the next 96 days.
2. If she is early there is a probability of 0.7 that she will eat a banana. If she is late she does not eat a banana. If she is on time there is a probability of 0.4 that she will eat a banana. Given that for one particular meeting with friends she does not eat a banana, find the probability that she is on time.

7 A bag contains three 1 p coins and seven 2 p coins. Coins are removed at random one at a time, without replacement, until the total value of the coins removed is at least 3p. Then no more coins are removed.

1. Copy and complete the probability tree diagram. First coin \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-4_350_317_1279_568} Find the probability that
2. exactly two coins are removed,
3. the total value of the coins removed is 4p.

6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\). The tree diagram shows the different combinations of three coloured discs chosen. \includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-5_863_986_559_612}

1. Write down the values of \(a , b , c , d , e\) and \(f\). The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).

   \(r\)	0	1	2	3
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 10 }\)	\(k\)	\(\frac { 9 } { 20 }\)	\(\frac { 1 } { 5 }\)

2. Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
3. Find the value of \(k\).
4. Calculate the mean and variance of \(R\).

2 A bag contains 5 black discs and 3 red discs. A disc is selected at random from the bag. If it is red it is replaced in the bag. If it is black, it is not replaced. A second disc is now selected at random from the bag. Find the probability that

1. the second disc is black, given that the first disc was black,
2. the second disc is black,
3. the two discs are of different colours.

4 A bag contains 6 white discs and 4 blue discs. Discs are removed at random, one at a time, without replacement.

1. Find the probability that
   1. the second disc is blue, given that the first disc was blue,
   2. the second disc is blue,
   3. the third disc is blue, given that the first disc was blue.
   4. The random variable \(X\) is the number of discs which are removed up to and including the first blue disc. State whether the variable X has a geometric distribution. Explain your answer briefly.

3 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{b35b2b3b-0d26-4a35-b4d2-110bf270d5dc-2_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 An amateur weather forecaster describes each day as either sunny, cloudy or wet. He keeps a record each day of his forecast and of the actual weather. His results for one particular year are given in the table. A day is selected at random from that year.

1. Show that the probability that the forecast is correct is \(\frac { 264 } { 365 }\). Find the probability that
2. the forecast is correct, given that the forecast is sunny,
3. the forecast is correct, given that the weather is wet,
4. the weather is cloudy, given that the forecast is correct.

5 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).

3 Steve is going on holiday. The probability that he is delayed on his outward flight is 0.3 . The probability that he is delayed on his return flight is 0.2 , independently of whether or not he is delayed on the outward flight.

1. Find the probability that Steve is delayed on his outward flight but not on his return flight.
2. Find the probability that he is delayed on at least one of the two flights.
3. Given that he is delayed on at least one flight, find the probability that he is delayed on both flights.

2 Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.

• \(A\) is the event that Isobel's parents watch a match.
• \(B\) is the event that Isobel scores in a match.

You are given that \(\mathrm { P } ( B \mid A ) = \frac { 3 } { 7 }\) and \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).

1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
4. By comparing \(\mathrm { P } ( B \mid A )\) with \(\mathrm { P } ( B )\), explain why Isobel should ask her parents not to attend.

6 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.
   
   [diagram]
   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.

4 A local council has introduced a recycling scheme for aluminium, paper and kitchen waste. 50 residents are asked which of these materials they recycle. The numbers of people who recycle each type of material are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-3_803_803_406_671} One of the residents is selected at random.

1. Find the probability that this resident recycles
   (A) at least one of the materials,
   (B) exactly one of the materials.
2. Given that the resident recycles aluminium, find the probability that this resident does not recycle paper. Two residents are selected at random.
3. Find the probability that exactly one of them recycles kitchen waste.

7 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, \(3 \%\) is classified as positive, \(6 \%\) as doubtful and the rest negative. In fact, of the people who test positive, only \(95 \%\) have the disease. Of the people who test doubtful, \(10 \%\) have the disease. Of the people who test negative, \(1 \%\) actually have the disease. People who do not have the disease are described as 'clear'.

1. Copy and complete the tree diagram to show this information. \includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-5_830_1157_845_536}
2. Find the probability that a randomly selected person tests negative and is clear.
3. Find the probability that a randomly selected person has the disease.
4. Find the probability that a randomly selected person tests negative given that the person has the disease.
5. Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in \(98 \%\) of cases. If a person is clear, the examination will always correctly identify this.
6. A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.

2 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.

• \(W\) is the event that this person speaks Welsh.
• \(C\) is the event that this person is a child.

You are given that \(\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17\) and \(\mathrm { P } ( W \cap C ) = 0.06\).

1. Determine whether the events \(W\) and \(C\) are independent.
2. Draw a Venn diagram, showing the events \(W\) and \(C\), and fill in the probability corresponding to each region of your diagram.
3. Find \(\mathrm { P } ( W \mid C )\).
4. Given that \(\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169\), use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults.

2 The box and whisker plot below summarises the weights in grams of the 20 chocolates in a box. \includegraphics[max width=\textwidth, alt={}, center]{452a52c9-b1fa-4b98-a85d-a34ba0f84a9d-1_290_1186_1099_452}

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.

1 A survey is being carried out into the sports viewing habits of people in a particular area. As part of the survey, 250 people are asked which of the following sports they have watched on television in the past month.

• Football
• Cycling
• Rugby

The numbers of people who have watched these sports are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{870b6ef1-60f7-42e3-95f8-0544a2a07b15-1_725_921_728_622} One of the people is selected at random.

1. Find the probability that this person has in the past month
   (A) watched cycling but not football,
   (B) watched either one or two of the three sports.
2. Given that this person has watched cycling, find the probability that this person has not watched football.

3 Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities \(0.2,0.5\) and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.

1. Draw a probability tree diagram to illustrate the outcomes.
2. Find the probability that a randomly selected candidate is accepted.
3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted.