2.03d Calculate conditional probability: from first principles

5 At a zoo, rides are offered on elephants, camels and jungle tractors. Ravi has money for only one ride. To decide which ride to choose, he tosses a fair coin twice. If he gets 2 heads he will go on the elephant ride, if he gets 2 tails he will go on the camel ride and if he gets 1 of each he will go on the jungle tractor ride.

1. Find the probabilities that he goes on each of the three rides. The probabilities that Ravi is frightened on each of the rides are as follows: $$\text { elephant ride } \frac { 6 } { 10 } , \quad \text { camel ride } \frac { 7 } { 10 } , \quad \text { jungle tractor ride } \frac { 8 } { 10 } .$$
2. Draw a fully labelled tree diagram showing the rides that Ravi could take and whether or not he is frightened. Ravi goes on a ride.
3. Find the probability that he is frightened.
4. Given that Ravi is not frightened, find the probability that he went on the camel ride.

7 In a television quiz show Peter answers questions one after another, stopping as soon as a question is answered wrongly.

• The probability that Peter gives the correct answer himself to any question is 0.7 .
• The probability that Peter gives a wrong answer himself to any question is 0.1 .
• The probability that Peter decides to ask for help for any question is 0.2 .

On the first occasion that Peter decides to ask for help he asks the audience. The probability that the audience gives the correct answer to any question is 0.95 . This information is shown in the tree diagram below. \includegraphics[max width=\textwidth, alt={}, center]{e7e0fcbe-ab96-4292-b3ad-c57b74f15301-3_394_649_1779_386} \includegraphics[max width=\textwidth, alt={}, center]{e7e0fcbe-ab96-4292-b3ad-c57b74f15301-3_270_743_2010_1023}

1. Show that the probability that the first question is answered correctly is 0.89 . On the second occasion that Peter decides to ask for help he phones a friend. The probability that his friend gives the correct answer to any question is 0.65 .
2. Find the probability that the first two questions are both answered correctly.
3. Given that the first two questions were both answered correctly, find the probability that Peter asked the audience.

3 Christa takes her dog for a walk every day. The probability that they go to the park on any day is 0.6 . If they go to the park there is a probability of 0.35 that the dog will bark. If they do not go to the park there is a probability of 0.75 that the dog will bark.

1. Find the probability that they go to the park on more than 5 of the next 7 days.
2. Find the probability that the dog barks on any particular day.
3. Find the variance of the number of times they go to the park in 30 days.

5 Set \(A\) consists of the ten digits \(0,0,0,0,0,0,2,2,2,4\).
Set \(B\) consists of the seven digits \(0,0,0,0,2,2,2\).
One digit is chosen at random from each set. The random variable \(X\) is defined as the sum of these two digits.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 3 } { 7 }\).
2. Tabulate the probability distribution of \(X\).
3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
4. Given that \(X = 2\), find the probability that the digit chosen from set \(A\) was 2 .

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.

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.

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.

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.

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.

3 A fair six-sided die is thrown twice and the scores are noted. Event \(X\) is defined as 'The total of the two scores is 4'. Event \(Y\) is defined as 'The first score is 2 or 5'. Are events \(X\) and \(Y\) independent? Justify your answer.

1 Two ordinary fair dice are thrown and the numbers obtained are noted. Event \(S\) is 'The sum of the numbers is even'. Event \(T\) is 'The sum of the numbers is either less than 6 or a multiple of 4 or both'. Showing your working, determine whether the events \(S\) and \(T\) are independent.

2 Megan sends messages to her friends in one of 3 different ways: text, email or social media. For each message, the probability that she uses text is 0.3 and the probability that she uses email is 0.2 . She receives an immediate reply from a text message with probability 0.4 , from an email with probability 0.15 and from social media with probability 0.6 .

1. Draw a fully labelled tree diagram to represent this information.
2. Given that Megan does not receive an immediate reply to a message, find the probability that the message was an email.

3 A fair eight-sided die has faces marked \(1,2,3,4,5,6,7,8\). The score when the die is thrown is the number on the face the die lands on. The die is thrown twice.

• Event \(R\) is 'one of the scores is exactly 3 greater than the other score'.
• Event \(S\) is 'the product of the scores is more than 19'.
  1. Find the probability of \(R\).
  2. Find the probability of \(S\).
  3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.

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 \(n\).

6 Pack \(A\) consists of ten cards numbered \(0,0,1,1,1,1,1,3,3,3\). Pack \(B\) consists of six cards numbered \(0,0,2,2,2,2\). One card is chosen at random from each pack. The random variable \(X\) is defined as the sum of the two numbers on the cards.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 2 } { 15 }\). \includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-08_59_1569_497_328}
2. Draw up the probability distribution table for \(X\).
3. Given that \(X = 3\), find the probability that the card chosen from pack \(A\) is a 1 .