Standard Bayes with discrete events

2 Georgie has a red scarf, a blue scarf and a yellow scarf. Each day she wears exactly one of these scarves. The probabilities for the three colours are \(0.2,0.45\) and 0.35 respectively. When she wears a red scarf, she always wears a hat. When she wears a blue scarf, she wears a hat with probability 0.4 . When she wears a yellow scarf, she wears a hat with probability 0.3 .

1. Find the probability that on a randomly chosen day Georgie wears a hat.
2. Find the probability that on a randomly chosen day Georgie wears a yellow scarf given that she does not wear a hat.

2 Jamie is equally likely to attend or not to attend a training session before a football match. If he attends, he is certain to be chosen for the team which plays in the match. If he does not attend, there is a probability of 0.6 that he is chosen for the team.

1. Find the probability that Jamie is chosen for the team.
2. Find the conditional probability that Jamie attended the training session, given that he was chosen for the team.

2 When Ted is looking for his pen, the probability that it is in his pencil case is 0.7 . If his pen is in his pencil case he always finds it. If his pen is somewhere else, the probability that he finds it is 0.2 . Given that Ted finds his pen when he is looking for it, find the probability that it was in his pencil case.

5

1. John plays two games of squash. The probability that he wins his first game is 0.3 . If he wins his first game, the probability that he wins his second game is 0.6 . If he loses his first game, the probability that he wins his second game is 0.15 . Given that he wins his second game, find the probability that he won his first game.
2. Jack has a pack of 15 cards. 10 cards have a picture of a robot on them and 5 cards have a picture of an aeroplane on them. Emma has a pack of cards. 7 cards have a picture of a robot on them and \(x - 3\) cards have a picture of an aeroplane on them. One card is taken at random from Jack's pack and one card is taken at random from Emma's pack. The probability that both cards have pictures of robots on them is \(\frac { 7 } { 18 }\). Write down an equation in terms of \(x\) and hence find the value of \(x\).

1 Ayman's breakfast drink is tea, coffee or hot chocolate with probabilities \(0.65,0.28,0.07\) respectively. When he drinks tea, the probability that he has milk in it is 0.8 . When he drinks coffee, the probability that he has milk in it is 0.5 . When he drinks hot chocolate he always has milk in it.

1. Draw a fully labelled tree diagram to represent this information.
2. Find the probability that Ayman's breakfast drink is coffee, given that his drink has milk in it.

7 During the school holidays, each day Khalid either rides on his bicycle with probability 0.6 , or on his skateboard with probability 0.4 . Khalid does not ride on both on the same day. If he rides on his bicycle then the probability that he hurts himself is 0.05 . If he rides on his skateboard the probability that he hurts himself is 0.75 .

1. Find the probability that Khalid hurts himself on any particular day.
2. Given that Khalid hurts himself on a particular day, find the probability that he is riding on his skateboard.
3. There are 45 days of school holidays. Show that the variance of the number of days Khalid rides on his skateboard is the same as the variance of the number of days that Khalid rides on his bicycle.
4. Find the probability that Khalid rides on his skateboard on at least 2 of 10 randomly chosen days in the school holidays.

5 In a certain town, 35\% of the people take a holiday abroad and 65\% take a holiday in their own country. Of those going abroad \(80 \%\) go to the seaside, \(15 \%\) go camping and \(5 \%\) take a city break. Of those taking a holiday in their own country, \(20 \%\) go to the seaside and the rest are divided equally between camping and a city break.

1. A person is chosen at random. Given that the person chosen goes camping, find the probability that the person goes abroad.
2. A group of \(n\) people is chosen randomly. The probability of all the people in the group taking a holiday in their own country is less than 0.002 . Find the smallest possible value of \(n\).

5 In a certain country \(54 \%\) of the population is male. It is known that \(5 \%\) of the males are colour-blind and \(2 \%\) of the females are colour-blind. A person is chosen at random and found to be colour-blind. By drawing a tree diagram, or otherwise, find the probability that this person is male.

3 When Andrea needs a taxi, she rings one of three taxi companies, A, B or C. 50\% of her calls are to taxi company \(A , 30 \%\) to \(B\) and \(20 \%\) to \(C\). A taxi from company \(A\) arrives late \(4 \%\) of the time, a taxi from company \(B\) arrives late \(6 \%\) of the time and a taxi from company \(C\) arrives late \(17 \%\) of the time.

1. Find the probability that, when Andrea rings for a taxi, it arrives late.
2. Given that Andrea's taxi arrives late, find the conditional probability that she rang company \(B\).

1 When Anya goes to school, the probability that she walks is 0.3 and the probability that she cycles is 0.65 ; if she does not walk or cycle she takes the bus. When Anya walks the probability that she is late is 0.15 . When she cycles the probability that she is late is 0.1 and when she takes the bus the probability that she is late is 0.6 . Given that Anya is late, find the probability that she cycles.

3 At the end of a revision course in mathematics, students have to pass a test to gain a certificate. The probability of any student passing the test at the first attempt is 0.85 . Those students who fail are allowed to retake the test once, and the probability of any student passing the retake test is 0.65 .

1. Draw a fully labelled tree diagram to show all the outcomes.
2. Given that a student gains the certificate, find the probability that this student fails the test on the first attempt.

7 Each question on a multiple-choice examination paper has \(n\) possible responses, only one of which is correct. Joni takes the paper and has probability \(p\), where \(0 < p < 1\), of knowing the correct response to any question, independently of any other. If she knows the correct response she will choose it, otherwise she will choose randomly from the \(n\) possibilities. The events \(K\) and \(A\) are 'Joni knows the correct response' and 'Joni answers correctly' respectively.

1. Show that \(\mathrm { P } ( A ) = \frac { q + n p } { n }\), where \(q = 1 - p\).
2. Find \(P ( K \mid A )\). A paper with 100 questions has \(n = 4\) and \(p = 0.5\). Each correct response scores 1 and each incorrect response scores - 1 .
3. (a) Joni answers all the questions on the paper and scores 40 . How many questions did she answer correctly?
   (b) By finding the distribution of the number of correct answers, or otherwise, find the probability that Joni scores at least 40 on the paper using her strategy.

3. For the events \(A\) and \(B\),

1. explain in words the meaning of the term \(\mathrm { P } \left( \begin{array} { l l } B & A \end{array} \right)\),
2. sketch a Venn diagram to illustrate the relationship \(\mathrm { P } \left( \begin{array} { l l } B & A \end{array} \right) = 0\). Three companies operate a bus service along a busy main road. Amber buses run \(50 \%\) of the service and \(2 \%\) of their buses are more than 5 minutes late. Blunder buses run \(30 \%\) of the service and \(10 \%\) of their buses are more than 5 minutes late. Clipper buses run the remainder of the service and only \(1 \%\) of their buses run more than 5 minutes late. Jean is waiting for a bus on the main road.
3. Find the probability that the first bus to arrive is an Amber bus that is more than 5 minutes late. Let \(A , B\) and \(C\) denote the events that Jean catches an Amber bus, a Blunder bus and a Clipper bus respectively. Let \(L\) denote the event that Jean catches a bus that is more than 5 minutes late.
4. Draw a Venn diagram to represent the events \(A , B , \mathrm { C }\) and \(L\). Calculate the probabilities associated with each region and write them in the appropriate places on the Venn diagram.
5. Find the probability that Jean catches a bus that is more than 5 minutes late.

4. A manufacturing plant produces electronic circuit boards that need to pass two quality checks - a mechanical inspection and an electrical test. Historical data shows that \(15 \%\) of boards fail the mechanical inspection. Of those that pass the mechanical inspection, \(8 \%\) fail the electrical test. Of those that fail the mechanical inspection, \(60 \%\) fail the electrical test.

1. If a board is randomly selected from production, what is the probability that it passes both inspections?
2. If a board is selected at random and is found to have passed the electrical test, what is the probability that it also passed the mechanical inspection?
3. The company continues to test boards from a large batch until finding one that passes both inspections. Each board is tested independently of all others. What is the probability that they need to test exactly 3 boards to find one that passes both inspections?
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