2.03c Conditional probability: using diagrams/tables

6 When Don plays tennis, \(65 \%\) of his first serves go into the correct area of the court. If the first serve goes into the correct area, his chance of winning the point is \(90 \%\). If his first serve does not go into the correct area, Don is allowed a second serve, and of these, \(80 \%\) go into the correct area. If the second serve goes into the correct area, his chance of winning the point is \(60 \%\). If neither serve goes into the correct area, Don loses the point.

1. Draw a tree diagram to represent this information.
2. Using your tree diagram, find the probability that Don loses the point.
3. Find the conditional probability that Don's first serve went into the correct area, given that he loses the point.

5 Data about employment for males and females in a small rural area are shown in the table. A person from this area is chosen at random. Let \(M\) be the event that the person is male and let \(E\) be the event that the person is employed.

1. Find \(\mathrm { P } ( M )\).
2. Find \(\mathrm { P } ( M\) and \(E )\).
3. Are \(M\) and \(E\) independent events? Justify your answer.
4. Given that the person chosen is unemployed, find the probability that the person is female.

2 The probability that Henk goes swimming on any day is 0.2 . On a day when he goes swimming, the probability that Henk has burgers for supper is 0.75 . On a day when he does not go swimming the probability that he has burgers for supper is \(x\). This information is shown on the following tree diagram. \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-2_693_1038_845_555} The probability that Henk has burgers for supper on any day is 0.5 .

1. Find \(x\).
2. Given that Henk has burgers for supper, find the probability that he went swimming that day.

5 Each father in a random sample of fathers was asked how old he was when his first child was born. The following histogram represents the information. \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_789_1627_1468_260}

1. What is the modal age group?
2. How many fathers were between 25 and 30 years old when their first child was born?
3. How many fathers were in the sample?
4. Find the probability that a father, chosen at random from the group, was between 25 and 30 years old when his first child was born, given that he was older than 25 years. 632 teams enter for a knockout competition, in which each match results in one team winning and the other team losing. After each match the winning team goes on to the next round, and the losing team takes no further part in the competition. Thus 16 teams play in the second round, 8 teams play in the third round, and so on, until 2 teams play in the final round.

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 In country \(A 30 \%\) of people who drink tea have sugar in it. In country \(B 65 \%\) of people who drink tea have sugar in it. There are 3 million people in country \(A\) who drink tea and 12 million people in country \(B\) who drink tea. A person is chosen at random from these 15 million people.

1. Find the probability that the person chosen is from country \(A\).
2. Find the probability that the person chosen does not have sugar in their tea.
3. Given that the person chosen does not have sugar in their tea, find the probability that the person is from country \(B\).

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.

7

1. 1. Find the probability of getting at least one 3 when 9 fair dice are thrown.
   2. When \(n\) fair dice are thrown, the probability of getting at least one 3 is greater than 0.9. Find the smallest possible value of \(n\).
2. A bag contains 5 green balls and 3 yellow balls. Ronnie and Julie play a game in which they take turns to draw a ball from the bag at random without replacement. The winner of the game is the first person to draw a yellow ball. Julie draws the first ball. Find the probability that Ronnie wins the game.

4 Tim throws a fair die twice and notes the number on each throw.

1. Tim calculates his final score as follows. If the number on the second throw is a 5 he multiplies the two numbers together, and if the number on the second throw is not a 5 he adds the two numbers together. Find the probability that his final score is
   1. 12,
   2. 5 .
   3. Events \(A , B , C\) are defined as follows. \(A\) : the number on the second throw is 5 \(B\) : the sum of the numbers is 6 \(C\) : the product of the numbers is even
      By calculation find which pairs, if any, of the events \(A , B\) and \(C\) are independent.

2 Maria has 3 pre-set stations on her radio. When she switches her radio on, there is a probability of 0.3 that it will be set to station 1, a probability of 0.45 that it will be set to station 2 and a probability of 0.25 that it will be set to station 3 . On station 1 the probability that the presenter is male is 0.1 , on station 2 the probability that the presenter is male is 0.85 and on station 3 the probability that the presenter is male is \(p\). When Maria switches on the radio, the probability that it is set to station 3 and the presenter is male is 0.075 .

1. Show that the value of \(p\) is 0.3 .
2. Given that Maria switches on and hears a male presenter, find the probability that the radio was set to station 2.

6 A box of biscuits contains 30 biscuits, some of which are wrapped in gold foil and some of which are unwrapped. Some of the biscuits are chocolate-covered. 12 biscuits are wrapped in gold foil, and of these biscuits, 7 are chocolate-covered. There are 17 chocolate-covered biscuits in total.

1. Copy and complete the table below to show the number of biscuits in each category.

   	Wrapped in gold foil	Unwrapped	Total
   Chocolate-covered
   Not chocolate-covered
   Total			30

   A biscuit is selected at random from the box.
2. Find the probability that the biscuit is wrapped in gold foil. The biscuit is returned to the box. An unwrapped biscuit is then selected at random from the box.
3. Find the probability that the biscuit is chocolate-covered. The biscuit is returned to the box. A biscuit is then selected at random from the box.
4. Find the probability that the biscuit is unwrapped, given that it is chocolate-covered. The biscuit is returned to the box. Nasir then takes 4 biscuits without replacement from the box.
5. Find the probability that he takes exactly 2 wrapped biscuits.

5 Suzanne has 20 pairs of shoes, some of which have designer labels. She has 6 pairs of high-heeled shoes, of which 2 pairs have designer labels. She has 4 pairs of low-heeled shoes, of which 1 pair has designer labels. The rest of her shoes are pairs of sports shoes. Suzanne has 8 pairs of shoes with designer labels in total.

1. Copy and complete the table below to show the number of pairs in each category.

   	Designer labels	No designer labels	Total
   High-heeled shoes
   Low-heeled shoes
   Sports shoes
   Total			20

   Suzanne chooses 1 pair of shoes at random to wear.
2. Find the probability that she wears the pair of low-heeled shoes with designer labels.
3. Find the probability that she wears a pair of sports shoes.
4. Find the probability that she wears a pair of high-heeled shoes, given that she wears a pair of shoes with designer labels.
5. State with a reason whether the events 'Suzanne wears a pair of shoes with designer labels' and 'Suzanne wears a pair of sports shoes' are independent. Suzanne chooses 1 pair of shoes at random each day.
6. Find the probability that Suzanne wears a pair of shoes with designer labels on at most 4 days out of the next 7 days.

7 Box \(A\) contains 8 white balls and 2 yellow balls. Box \(B\) contains 5 white balls and \(x\) yellow balls. A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\). The tree diagram below shows the possibilities for the colours of the balls chosen. \begin{figure}[h] \end{figure}

1. Justify the probability \(\frac { x } { x + 6 }\) on the tree diagram.
2. Copy and complete the tree diagram.
3. If the ball chosen from box \(A\) is white then the probability that the ball chosen from box \(B\) is also white is \(\frac { 1 } { 3 }\). Show that the value of \(x\) is 12 .
4. Given that the ball chosen from box \(B\) is yellow, find the conditional probability that the ball chosen from box \(A\) was yellow.

7 Susan has a bag of sweets containing 7 chocolates and 5 toffees. Ahmad has a bag of sweets containing 3 chocolates, 4 toffees and 2 boiled sweets. A sweet is taken at random from Susan's bag and put in Ahmad's bag. A sweet is then taken at random from Ahmad's bag.

1. Find the probability that the two sweets taken are a toffee from Susan's bag and a boiled sweet from Ahmad's bag.
2. Given that the sweet taken from Ahmad's bag is a chocolate, find the probability that the sweet taken from Susan's bag was also a chocolate.
3. The random variable \(X\) is the number of times a chocolate is taken. State the possible values of \(X\) and draw up a table to show the probability distribution of \(X\).

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 \(x\).

5 Playground equipment consists of swings ( \(S\) ), roundabouts ( \(R\) ), climbing frames ( \(C\) ) and play-houses \(( P )\). The numbers of pieces of equipment in each of 3 playgrounds are as follows. Each day Nur takes her child to one of the playgrounds. The probability that she chooses playground \(X\) is \(\frac { 1 } { 4 }\). The probability that she chooses playground \(Y\) is \(\frac { 1 } { 4 }\). The probability that she chooses playground \(Z\) is \(\frac { 1 } { 2 }\). When she arrives at the playground, she chooses one piece of equipment at random.

1. Find the probability that Nur chooses a play-house.
2. Given that Nur chooses a climbing frame, find the probability that she chose playground \(Y\).

3 Roger and Andy play a tennis match in which the first person to win two sets wins the match. The probability that Roger wins the first set is 0.6 . For sets after the first, the probability that Roger wins the set is 0.7 if he won the previous set, and is 0.25 if he lost the previous set. No set is drawn.

1. Find the probability that there is a winner of the match after exactly two sets.
2. Find the probability that Andy wins the match given that there is a winner of the match after exactly two sets.

6 Tom and Ben play a game repeatedly. The probability that Tom wins any game is 0.3 . Each game is won by either Tom or Ben. Tom and Ben stop playing when one of them (to be called the champion) has won two games.

1. Find the probability that Ben becomes the champion after playing exactly 2 games.
2. Find the probability that Ben becomes the champion.
3. Given that Tom becomes the champion, find the probability that he won the 2nd game.

3 Jason throws two fair dice, each with faces numbered 1 to 6 . Event \(A\) is 'one of the numbers obtained is divisible by 3 and the other number is not divisible by 3 '. Event \(B\) is 'the product of the two numbers obtained is even'.

1. Determine whether events \(A\) and \(B\) are independent, showing your working.
2. Are events \(A\) and \(B\) mutually exclusive? Justify your answer.

4 \includegraphics[max width=\textwidth, alt={}, center]{c0c7e038-805a-4237-a579-a6571b84f337-2_451_1530_1393_303} A survey is undertaken to investigate how many photos people take on a one-week holiday and also how many times they view past photos. For a randomly chosen person, the probability of taking fewer than 100 photos is \(x\). The probability that these people view past photos at least 3 times is 0.76 . For those who take at least 100 photos, the probability that they view past photos fewer than 3 times is 0.90 . This information is shown in the tree diagram. The probability that a randomly chosen person views past photos fewer than 3 times is 0.801 .

1. Find \(x\).
2. Given that a person views past photos at least 3 times, find the probability that this person takes at least 100 photos.

2 When Joanna cooks, the probability that the meal is served on time is \(\frac { 1 } { 5 }\). The probability that the kitchen is left in a mess is \(\frac { 3 } { 5 }\). The probability that the meal is not served on time and the kitchen is not left in a mess is \(\frac { 3 } { 10 }\). Some of this information is shown in the following table.

1. Copy and complete the table.
2. Given that the kitchen is left in a mess, find the probability that the meal is not served on time.

3 The probability that the school bus is on time on any particular day is 0.6 . If the bus is on time the probability that Sam the driver gets a cup of coffee is 0.9 . If the bus is not on time the probability that Sam gets a cup of coffee is 0.3 .

1. Find the probability that Sam gets a cup of coffee.
2. Given that Sam does not get a cup of coffee, find the probability that the bus is not on time.

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.

1 In a group of 30 adults, 25 are right-handed and 8 wear spectacles. The number who are right-handed and do not wear spectacles is 19 .

1. Copy and complete the following table to show the number of adults in each category.

   	Wears spectacles	Does not wear spectacles	Total
   Right-handed
   Not right-handed
   Total			30

   An adult is chosen at random from the group. Event \(X\) is 'the adult chosen is right-handed'; event \(Y\) is 'the adult chosen wears spectacles'.
2. Determine whether \(X\) and \(Y\) are independent events, justifying your answer.

2 Ashfaq throws two fair dice and notes the numbers obtained. \(R\) is the event 'The product of the two numbers is 12 '. \(T\) is the event 'One of the numbers is odd and one of the numbers is even'. By finding appropriate probabilities, determine whether events \(R\) and \(T\) are independent.