2.03a Mutually exclusive and independent events

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.

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.

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.

6 In a large town, 79\% of the population were born in England, 20\% in the rest of the UK and the remaining 1\% overseas. Two people are selected at random. You may use the tree diagram below in answering this question. \includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-4_946_1119_580_513}

1. Find the probability that
   (A) both of these people were born in the rest of the UK,
   (B) at least one of these people was born in England,
   (C) neither of these people was born overseas.
2. Find the probability that both of these people were born in the rest of the UK given that neither was born overseas.
3. (A) Five people are selected at random. Find the probability that at least one of them was not born in England.
   (B) An interviewer selects \(n\) people at random. The interviewer wishes to ensure that the probability that at least one of them was not born in England is more than \(90 \%\). Find the least possible value of \(n\). You must show working to justify your answer.

2 An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.

1. Three days are chosen at random. Find the probability that the pollution level is
   (A) low on all 3 days,
   (B) low on at least one day,
   (C) low on one day, medium on another day, and high on the other day.
2. Ten days are chosen at random. Find the probability that
   (A) there are no days when the pollution level is high,
   (B) there is exactly one day when the pollution level is high. The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
3. Carry out a test at the \(5 \%\) level to determine if there is evidence to suggest that she is correct. Use hypotheses \(\mathrm { H } _ { 0 } : p = 0.5 , \mathrm { H } _ { 1 } : p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(\mathrm { H } _ { 1 }\) has this form.

2 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.

1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
   (A) both seeds germinate,
   (B) just one seed germinates,
   (C) neither seed germinates.
2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.

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.

3 The birth weights of 200 lambs from crossbred sheep are illustrated by the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_919_1144_430_476}

1. Estimate the percentage of lambs with birth weight over 6 kg .
2. Estimate the median and interquartile range of the data.
3. Use your answers to part (ii) to show that there are very few, if any, outliers. Comment briefly on whether any outliers should be disregarded in analysing these data. The box and whisker plot shows the birth weights of 100 lambs from Welsh Mountain sheep. \includegraphics[max width=\textwidth, alt={}, center]{ab4d5ab1-e3b7-495f-9142-d37df7e712de-3_321_1610_1818_293}
4. Use appropriate measures to compare briefly the central tendencies and variations of the weights of the two types of lamb.
5. The weight of the largest Welsh Mountain lamb was originally recorded as 6.5 kg , but then corrected. If this error had not been corrected, how would this have affected your answers to part (iv)? Briefly explain your answer.
6. One lamb of each type is selected at random. Estimate the probability that the birth weight of both lambs is at least 3.9 kg .

3 The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

1. Draw a cumulative frequency diagram to illustrate these data.
2. Use the diagram to estimate the median and interquartile range of the data.
3. Show that there are no outliers in the sample.
4. Three of these 50 sacks are selected at random. Find the probability that
   (A) in all three, the heating quality \(x\) is more than 9.5 , \(( B )\) in at least two, the heating quality \(x\) is more than 9.5.

2 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.

1. Find the probability that neither card is an ace.
2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.

4 Each weekday, Marta travels to school by bus. Sometimes she arrives late.

• \(L\) is the event that Marta arrives late.
• \(R\) is the event that it is raining.

You are given that \(\mathrm { P } ( L ) = 0.15 , \mathrm { P } ( R ) = 0.22\) and \(\mathrm { P } ( L \mid R ) = 0.45\).

1. Use this information to show that the events \(L\) and \(R\) are not independent.
2. Find \(\mathrm { P } ( L \cap R )\).
3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.

7 At a garden centre there is a box containing 50 hyacinth bulbs. Of these, 30 will produce a blue flower and the remaining 20 will produce a red flower. Unfortunately they have become mixed together so that it is not known which of the bulbs will produce a blue flower and which will produce a red flower. Karen buys 3 of these bulbs.

1. Find the probability that all 3 of these bulbs will produce blue flowers.
2. Find the probability that Karen will have at least one flower of each colour from her 3 bulbs.

2 Each packet of Cruncho cereal contains one free fridge magnet. There are five different types of fridge magnet to collect. They are distributed, with equal probability, randomly and independently in the packets. Keith is about to start collecting these fridge magnets.

1. Find the probability that the first 2 packets that Keith buys contain the same type of fridge magnet.
2. Find the probability that Keith collects all five types of fridge magnet by buying just 5 packets.
3. Hence find the probability that Keith has to buy more than 5 packets to acquire a complete set.

4 In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random.

• \(G\) is the event that this person goes to the gym.
• \(R\) is the event that this person goes running.

You are given that \(\mathrm { P } ( G ) = 0.24 , \mathrm { P } ( R ) = 0.13\) and \(\mathrm { P } ( G \cap R ) = 0.06\).

1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
2. Determine whether the events \(G\) and \(R\) are independent.
3. Find \(\mathrm { P } ( R \mid G )\).

5 My credit card has a 4-digit code called a PIN. You should assume that any 4-digit number from 0000 to 9999 can be a PIN.

1. If I cannot remember any digits and guess my number, find the probability that I guess it correctly. In fact my PIN consists of four different digits. I can remember all four digits, but cannot remember the correct order.
2. If I now guess my number, find the probability that I guess it correctly.

2 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.

3 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. [1]

1 In a large town, 79\% of the population were born in England, 20\% in the rest of the UK and the remaining \(1 \%\) overseas. Two people are selected at random. You may use the tree diagram below in answering this question. \includegraphics[max width=\textwidth, alt={}, center]{b56ccabe-0e51-4555-b550-78ba347f69bb-1_944_1118_626_547}

1. Find the probability that
   (A) both of these people were born in the rest of the UK,
   (B) at least one of these people was born in England,
   (C) neither of these people was born overseas.
2. Find the probability that both of these people were born in the rest of the UK given that neither was born overseas.
3. (A) Five people are selected at random. Find the probability that at least one of them was not born in England.
   (B) An interviewer selects \(n\) people at random. The interviewer wishes to ensure that the probability that at least one of them was not born in England is more than \(90 \%\). Find the least possible value of \(n\). You must show working to justify your answer.

2 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.

3 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.
• \(\quad B\) is the event that Isobel scores in a match.

You are given that \(\frac { 3 } { 7 }\) 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.

1 A school athletics team has 10 members. The table shows which competitions each of the members can take part in. An athlete is selected at random. Events \(A , B , C , D\) are defined as follows. \(A\) : the athlete can take part in exactly 2 competitions. \(B\) : the athlete can take part in the 200 m . \(C\) : the athlete can take part in the 110 m hurdles. \(D\) : the athlete can take part in the long jump.

1. Write down the value of \(\mathrm { P } ( A \cap B )\).
2. Write down the value of \(\mathrm { P } ( C \cup D )\).
3. Which two of the four events \(A , B , C , D\) are mutually exclusive?
4. Show that events \(B\) and \(D\) are not independent.

5 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-5_480_771_452_655} 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 \(\mathrm { P } ( B )\).

4 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.

1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
   (A) both seeds germinate,
   (B) just one seed germinates,
   (C) neither seed germinates.
2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.

6 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a \(\pounds 10\) prize, 20 of them have a \(\pounds 100\) prize, one of them has a \(\pounds 5000\) prize and all of the rest have no prize. This information is summarised in the frequency table below.

1. Find the mean and standard deviation of the prize money per ticket.
2. I buy two of these tickets at random. Find the probability that I win either two \(\pounds 10\) prizes or two \(\pounds 100\) prizes.

3 Sixty people each make two throws with a fair six-sided die.

1. State the probability of one particular person obtaining two sixes.
2. Using a suitable approximation, calculate the probability that at least four of the sixty obtain two sixes.