Population partition tree diagram

5 On Mondays, Rani cooks her evening meal. She has a pizza, a burger or a curry with probabilities \(0.35,0.44,0.21\) respectively. When she cooks a pizza, Rani has some fruit with probability 0.3 . When she cooks a burger, she has some fruit with probability 0.8 . When she cooks a curry, she never has any fruit.

1. Draw a fully labelled tree diagram to represent this information.
2. Find the probability that Rani has some fruit.
3. Find the probability that Rani does not have a burger given that she does not have any fruit.

5 Each day John either cycles to work or goes on the bus.

• If it is raining when John is ready to set off for work, the probability that he cycles to work is 0.4.
• If it is not raining when John is ready to set off for work, the probability that he cycles to work is 0.9 .
• The probability that it is raining when he is ready to set off for work is 0.2 .

You should assume that days on which it rains occur randomly and independently.

1. Draw a tree diagram to show the possible outcomes and their associated probabilities.
2. Calculate the probability that, on a randomly chosen day, John cycles to work. John works 5 days each week.
3. Calculate the probability that he cycles to work every day in a randomly chosen working week.

2. On a randomly chosen day the probability that Bill travels to school by car, by bicycle or on foot is \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. The probability of being late when using these methods of travel is \(\frac { 1 } { 5 } , \frac { 2 } { 5 }\) and \(\frac { 1 } { 10 }\) respectively.

1. Draw a tree diagram to represent this information.
2. Find the probability that on a randomly chosen day
   1. Bill travels by foot and is late,
   2. Bill is not late.
3. Given that Bill is late, find the probability that he did not travel on foot.

7. In a large company, 78\% of employees are car owners,
\(30 \%\) of these car owners are also bike owners,
85\% of those who are not car owners are bike owners.

1. Draw a tree diagram to represent this information. An employee is selected at random.
2. Find the probability that the employee is a car owner or a bike owner but not both. Another employee is selected at random. Given that this employee is a bike owner,
3. find the probability that the employee is a car owner. Two employees are selected at random.
4. Find the probability that only one of them is a bike owner.

1. In a factory, three machines, \(J , K\) and \(L\), are used to make biscuits.

Machine \(J\) makes \(25 \%\) of the biscuits. Machine \(K\) makes \(45 \%\) of the biscuits. The rest of the biscuits are made by machine \(L\).
It is known that \(2 \%\) of the biscuits made by machine \(J\) are broken, \(3 \%\) of the biscuits made by machine \(K\) are broken and 5\% of the biscuits made by machine \(L\) are broken.

1. Draw a tree diagram to illustrate all the possible outcomes and associated probabilities. A biscuit is selected at random.
2. Calculate the probability that the biscuit is made by machine \(J\) and is not broken.
3. Calculate the probability that the biscuit is broken.
4. Given that the biscuit is broken, find the probability that it was not made by machine \(K\).

5. A keep-fit enthusiast swims, runs or cycles each day with probabilities \(0.2,0.3\) and 0.5 respectively. If he swims he then spends time in the sauna with probability 0.35 . The probabilities that he spends time in the sauna after running or cycling are 0.2 and 0.45 respectively.

1. Represent this information on a tree diagram.
2. Find the probability that on any particular day he uses the sauna.
3. Given that he uses the sauna one day, find the probability that he had been swimming.
4. Given that he did not use the sauna one day, find the probability that he had been swimming.

4 A manufacturer produces three models of washing machine: basic, standard and deluxe. An analysis of warranty records shows that \(25 \%\) of faults are on basic machines, \(60 \%\) are on standard machines and 15\% are on deluxe machines. For basic machines, 30\% of faults reported during the warranty period are electrical, \(50 \%\) are mechanical and \(20 \%\) are water-related. For standard machines, 40\% of faults reported during the warranty period are electrical, \(45 \%\) are mechanical and 15\% are water-related. For deluxe machines, \(55 \%\) of faults reported during the warranty period are electrical, \(35 \%\) are mechanical and \(10 \%\) are water-related.

1. Draw a tree diagram to represent the above information.
2. Hence, or otherwise, determine the probability that a fault reported during the warranty period:
   1. is electrical;
   2. is on a deluxe machine, given that it is electrical.
3. A random sample of 10 electrical faults reported during the warranty period is selected. Calculate the probability that exactly 4 of them are on deluxe machines.

4 It is proposed to introduce, for all males at age 60, screening tests, A and B, for a certain disease. Test B is administered only when the result of Test A is inconclusive. It is known that 10\% of 60-year-old men suffer from the disease. For those 60 -year-old men suffering from the disease:

• Test A is known to give a positive result, indicating a presence of the disease, in \(90 \%\) of cases, a negative result in \(2 \%\) of cases and a requirement for the administration of Test B in \(8 \%\) of cases;
• Test B is known to give a positive result in \(98 \%\) of cases and a negative result in 2\% of cases.

For those 60 -year-old men not suffering from the disease:

• Test A is known to give a positive result in \(1 \%\) of cases, a negative result in \(80 \%\) of cases and a requirement for the administration of Test B in 19\% of cases;
• Test B is known to give a positive result in \(1 \%\) of cases and a negative result in \(99 \%\) of cases.
  1. Draw a tree diagram to represent the above information.
  2. 1. Hence, or otherwise, determine the probability that:
        (A) a 60-year-old man, suffering from the disease, tests negative;
        (B) a 60-year-old man, not suffering from the disease, tests positive.
     2. A random sample of ten thousand 60-year-old men is given the screening tests. Calculate, to the nearest 10, the number who you would expect to be given an incorrect diagnosis.
  3. Determine the probability that:
     1. a 60-year-old man suffers from the disease given that the tests provide a positive result;
     2. a 60-year-old man does not suffer from the disease given that the tests provide a negative result.

2 On a rail route between two stations, A and \(\mathrm { B } , 90 \%\) of trains leave A on time and \(10 \%\) of trains leave A late. Of those trains that leave A on time, \(15 \%\) arrive at B early, \(75 \%\) arrive on time and \(10 \%\) arrive late. Of those trains that leave A late, \(35 \%\) arrive at B on time and \(65 \%\) arrive late.

1. Represent this information by a fully-labelled tree diagram.
2. Hence, or otherwise, calculate the probability that a train:
   1. arrives at B early or on time;
   2. left A on time, given that it arrived at B on time;
   3. left A late, given that it was not late in arriving at B .
3. Two trains arrive late at B. Assuming that their journey times are independent, calculate the probability that exactly one train left A on time.

3 An investigation was carried out into the type of vehicle being driven when its driver was caught speeding. The investigation was restricted to drivers who were caught speeding when driving vehicles with at least 4 wheels. An analysis of the results showed that \(65 \%\) were driving cars ( C ), \(20 \%\) were driving vans (V) and 15\% were driving lorries (L). Of those driving cars, \(30 \%\) were caught by fixed speed cameras (F), 55\% were caught by mobile speed cameras (M) and 15\% were caught by average speed cameras (A). Of those driving vans, \(35 \%\) were caught by fixed speed cameras (F), \(45 \%\) were caught by mobile speed cameras (M) and 20\% were caught by average speed cameras (A). Of those driving lorries, \(10 \%\) were caught by fixed speed cameras \(( \mathrm { F } )\), \(65 \%\) were caught by mobile speed cameras (M) and \(25 \%\) were caught by average speed cameras (A).

1. Represent this information by a tree diagram on which are shown labels and percentages or probabilities.
2. Hence, or otherwise, calculate the probability that a driver, selected at random from those caught speeding:
   1. was driving either a car or a lorry and was caught by a mobile speed camera;
   2. was driving a lorry, given that the driver was caught by an average speed camera;
   3. was not caught by a fixed speed camera, given that the driver was not driving a car.
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3. Three drivers were selected at random from those caught speeding by fixed speed cameras. Calculate the probability that they were driving three different types of vehicle.
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3. A group of students were surveyed by a principal and \(\frac { 2 } { 3 }\) were found to always hand in assignments on time. When questioned about their assignments \(\frac { 3 } { 5 }\) said they always start their assignments on the day they are issued and, of those who always start their assignments on the day they are issued, \(\frac { 11 } { 20 }\) hand them in on time.
a Draw a tree diagram to represent this information.
b Find the probability that a randomly selected student:
i always start their assignments on the day they are issued and hand them in on time.
ii does not always hand in assignments on time and does not start their assignments on the day they are issued.
c Determine whether or not always starting assignments on the day they are issued and handing them in on time are statistically independent. Give reasons for your answer.
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4. A company assembles drills using components from two sources. Goodbuy supplies the components for \(85 \%\) of the drills whilst Amart supplies the components for the rest.
It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.
a Represent this information on a tree diagram. An assembled drill is selected at random.
b Find the probability that the drill is not faulty.