2.03c Conditional probability: using diagrams/tables

3 Redbury United soccer team play a match every week. Each match can be won, drawn or lost. At the beginning of the soccer season the probability that Redbury United win their first match is \(\frac { 3 } { 5 }\), with equal probabilities of losing or drawing. If they win the first match, the probability that they win the second match is \(\frac { 7 } { 10 }\) and the probability that they lose the second match is \(\frac { 1 } { 10 }\). If they draw the first match they are equally likely to win, draw or lose the second match. If they lose the first match, the probability that they win the second match is \(\frac { 3 } { 10 }\) and the probability that they draw the second match is \(\frac { 1 } { 20 }\).

1. Draw a fully labelled tree diagram to represent the first two matches played by Redbury United in the soccer season.
2. Given that Redbury United win the second match, find the probability that they lose the first match.

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.

3 A shop sells two makes of coffee, Café Premium and Café Standard. Both coffees come in two sizes, large jars and small jars. Of the jars on sale, \(65 \%\) are Café Premium and \(35 \%\) are Café Standard. Of the Café Premium, 40\% of the jars are large and of the Café Standard, 25\% of the jars are large. A jar is chosen at random.

1. Find the probability that the jar is small.
2. Find the probability that the jar is Café Standard given that it is large.

2 In a group of students, \(\frac { 3 } { 4 }\) are male. The proportion of male students who like their curry hot is \(\frac { 3 } { 5 }\) and the proportion of female students who like their curry hot is \(\frac { 4 } { 5 }\). One student is chosen at random.

1. Find the probability that the student chosen is either female, or likes their curry hot, or is both female and likes their curry hot.
2. Showing your working, determine whether the events 'the student chosen is male' and 'the student chosen likes their curry hot' are independent.

3 The members of a swimming club are classified either as 'Advanced swimmers' or 'Beginners'. The proportion of members who are male is \(x\), and the proportion of males who are Beginners is 0.7 . The proportion of females who are Advanced swimmers is 0.55 . This information is shown in the tree diagram. \includegraphics[max width=\textwidth, alt={}, center]{dd75fa20-fead-48d6-aff4-c5e733769f9f-04_435_974_482_587} For a randomly chosen member, the probability of being an Advanced swimmer is the same as the probability of being a Beginner.

1. Find \(x\).
2. Given that a randomly chosen member is an Advanced swimmer, find the probability that the member is male.

5 Rachel and Anna play each other at badminton. Each game results in either a win for Rachel or a win for Anna. The probability of Rachel winning the first game is 0.6 . If Rachel wins a particular game, the probability of her winning the next game is 0.7 , but if she loses, the probability of her winning the next game is 0.4 . By using a tree diagram, or otherwise,

1. find the conditional probability that Rachel wins the first game, given that she loses the second,
2. find the probability that Rachel wins 2 games and loses 1 game out of the first three games they play.

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 \(B\).

2 Boxes of sweets contain toffees and chocolates. Box \(A\) contains 6 toffees and 4 chocolates, box \(B\) contains 5 toffees and 3 chocolates, and box \(C\) contains 3 toffees and 7 chocolates. One of the boxes is chosen at random and two sweets are taken out, one after the other, and eaten.

1. Find the probability that they are both toffees.
2. Given that they are both toffees, find the probability that they both came from box \(A\).

7 Box \(A\) contains 5 red paper clips and 1 white paper clip. Box \(B\) contains 7 red paper clips and 2 white paper clips. One paper clip is taken at random from box \(A\) and transferred to box \(B\). One paper clip is then taken at random from box \(B\).

1. Find the probability of taking both a white paper clip from box \(A\) and a red paper clip from box \(B\).
2. Find the probability that the paper clip taken from box \(B\) is red.
3. Find the probability that the paper clip taken from box \(A\) was red, given that the paper clip taken from box \(B\) is red.
4. The random variable \(X\) denotes the number of times that a red paper clip is taken. Draw up a table to show the probability distribution of \(X\).

6 There are three sets of traffic lights on Karinne's journey to work. The independent probabilities that Karinne has to stop at the first, second and third set of lights are \(0.4,0.8\) and 0.3 respectively.

1. Draw a tree diagram to show this information.
2. Find the probability that Karinne has to stop at each of the first two sets of lights but does not have to stop at the third set.
3. Find the probability that Karinne has to stop at exactly two of the three sets of lights.
4. Find the probability that Karinne has to stop at the first set of lights, given that she has to stop at exactly two sets of lights.

6 A box contains 4 pears and 7 oranges. Three fruits are taken out at random and eaten. Find the probability that

1. 2 pears and 1 orange are eaten, in any order,
2. the third fruit eaten is an orange,
3. the first fruit eaten was a pear, given that the third fruit eaten is an orange. There are 121 similar boxes in a warehouse. One fruit is taken at random from each box.
4. Using a suitable approximation, find the probability that fewer than 39 are pears.

3 Maria chooses toast for her breakfast with probability 0.85 . If she does not choose toast then she has a bread roll. If she chooses toast then the probability that she will have jam on it is 0.8 . If she has a bread roll then the probability that she will have jam on it is 0.4 .

1. Draw a fully labelled tree diagram to show this information.
2. Given that Maria did not have jam for breakfast, find the probability that she had toast.

7 James has a fair coin and a fair tetrahedral die with four faces numbered 1, 2, 3, 4. He tosses the coin once and the die twice. The random variable \(X\) is defined as follows.

• If the coin shows a head then \(X\) is the sum of the scores on the two throws of the die.
• If the coin shows a tail then \(X\) is the score on the first throw of the die only.
  1. Explain why \(X = 1\) can only be obtained by throwing a tail, and show that \(\mathrm { P } ( X = 1 ) = \frac { 1 } { 8 }\).
  2. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 16 }\).
  3. Copy and complete the probability distribution table for \(X\).

Event \(Q\) is 'James throws a tail'. Event \(R\) is 'the value of \(X\) is 7'.

Determine whether events \(Q\) and \(R\) are exclusive. Justify your answer.

2 On Saturday afternoons Mohit goes shopping with probability 0.25, or goes to the cinema with probability 0.35 or stays at home. If he goes shopping the probability that he spends more than \(\\) 50\( is 0.7 . If he goes to the cinema the probability that he spends more than \)\\( 50\) is 0.8 . If he stays at home he spends \(\\) 10$ on a pizza.

1. Find the probability that Mohit will go to the cinema and spend less than \(\\) 50\(.
2. Given that he spends less than \)\\( 50\), find the probability that he went to the cinema.

7 Dayo chooses two digits at random, without replacement, from the 9-digit number 113333555.

1. Find the probability that the two digits chosen are equal.
2. Find the probability that one digit is a 5 and one digit is not a 5 .
3. Find the probability that the first digit Dayo chose was a 5, given that the second digit he chose is not a 5 .
4. The random variable \(X\) is the number of 5s that Dayo chooses. Draw up a table to show the probability distribution of \(X\).

3 Jodie tosses a biased coin and throws two fair tetrahedral dice. The probability that the coin shows a head is \(\frac { 1 } { 3 }\). Each of the dice has four faces, numbered \(1,2,3\) and 4 . Jodie's score is calculated from the numbers on the faces that the dice land on, as follows:

• if the coin shows a head, the two numbers from the dice are added together;
• if the coin shows a tail, the two numbers from the dice are multiplied together.

Find the probability that the coin shows a head given that Jodie's score is 8 .

7 A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable \(X\) is the number of apples which have been taken when the process stops.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 3 }\).
2. Draw up the probability distribution table for \(X\). Another box contains 2 yellow peppers and 5 orange peppers. Three peppers are taken at random from the box without replacement.
3. Given that at least 2 of the peppers taken from the box are orange, find the probability that all 3 peppers are orange.

2 In country \(X , 25 \%\) of people have fair hair. In country \(Y , 60 \%\) of people have fair hair. There are 20 million people in country \(X\) and 8 million people in country \(Y\). A person is chosen at random from these 28 million people.

1. Find the probability that the person chosen is from country \(X\).
2. Find the probability that the person chosen has fair hair.
3. Find the probability that the person chosen is from country \(X\), given that the person has fair hair.

6 Deeti has 3 red pens and 1 blue pen in her left pocket and 3 red pens and 1 blue pen in her right pocket. 'Operation \(T\) ' consists of Deeti taking one pen at random from her left pocket and placing it in her right pocket, then taking one pen at random from her right pocket and placing it in her left pocket.

1. Find the probability that, when Deeti carries out operation \(T\), she takes a blue pen from her left pocket and then a blue pen from her right pocket. The random variable \(X\) is the number of blue pens in Deeti's left pocket after carrying out operation \(T\).
2. Find \(\mathrm { P } ( X = 1 )\).
3. Given that the pen taken from Deeti's right pocket is blue, find the probability that the pen taken from Deeti's left pocket is blue.

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.

5 Over a period of time Julian finds that on long-distance flights he flies economy class on \(82 \%\) of flights. On the rest of the flights he flies first class. When he flies economy class, the probability that he gets a good night's sleep is \(x\). When he flies first class, the probability that he gets a good night's sleep is 0.9 .

1. Draw a fully labelled tree diagram to illustrate this situation. The probability that Julian gets a good night's sleep on a randomly chosen flight is 0.285 .
2. Find the value of \(x\).
3. Given that on a particular flight Julian does not get a good night's sleep, find the probability that he is flying economy class.

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 In a group of students, the numbers of boys and girls studying Art, Music and Drama are given in the following table. Each of these 160 students is studying exactly one of these subjects.

1. Find the probability that a randomly chosen student is studying Music.
2. Determine whether the events 'a randomly chosen student is a boy' and 'a randomly chosen student is studying Music' are independent, justifying your answer.
3. Find the probability that a randomly chosen student is not studying Drama, given that the student is a girl.
4. Three students are chosen at random. Find the probability that exactly 1 is studying Music and exactly 2 are boys.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 Jake attempts the crossword puzzle in his daily newspaper every day. The probability that he will complete the puzzle on any given day is 0.75 , independently of all other days.

1. Find the probability that he will complete the puzzle at least three times over a period of five days.
   Kenny also attempts the puzzle every day. The probability that he will complete the puzzle on a Monday is 0.8 . The probability that he will complete it on a Tuesday is 0.9 if he completed it on the previous day and 0.6 if he did not complete it on the previous day.
2. Find the probability that Kenny will complete the puzzle on at least one of the two days Monday and Tuesday in a randomly chosen week.