2.03a Mutually exclusive and independent events

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.

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.

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 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 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.

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.

4 Two identical biased triangular spinners with sides marked 1,2 and 3 are spun. For each spinner, the probabilities of landing on the sides marked 1,2 and 3 are \(p , q\) and \(r\) respectively. The score is the sum of the numbers on the sides on which the spinners land. You are given that \(\mathrm { P } (\) score is \(6 ) = \frac { 1 } { 36 }\) and \(\mathrm { P } (\) score is \(5 ) = \frac { 1 } { 9 }\). Find the values of \(p , q\) and \(r\).

1 A biased die has faces numbered 1 to 6 . The probabilities of the die landing on 1,3 or 5 are each equal to 0.1 . The probabilities of the die landing on 2 or 4 are each equal to 0.2 . The die is thrown twice. Find the probability that the sum of the numbers it lands on is 9 .

6 Vehicles approaching a certain road junction from town \(A\) can either turn left, turn right or go straight on. Over time it has been noted that of the vehicles approaching this particular junction from town \(A\), \(55 \%\) turn left, \(15 \%\) turn right and \(30 \%\) go straight on. The direction a vehicle takes at the junction is independent of the direction any other vehicle takes at the junction.

1. Find the probability that, of the next three vehicles approaching the junction from town \(A\), one goes straight on and the other two either both turn left or both turn right.
2. Three vehicles approach the junction from town \(A\). Given that all three drivers choose the same direction at the junction, find the probability that they all go straight on.

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.

2 Jameel has 5 plums and 3 apricots in a box. Rosa has \(x\) plums and 6 apricots in a box. One fruit is chosen at random from Jameel's box and one fruit is chosen at random from Rosa's box. The probability that both fruits chosen are plums is \(\frac { 1 } { 4 }\). Write down an equation in \(x\) and hence find \(x\). [3]

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.

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.

2 Ivan throws three fair dice.

1. List all the possible scores on the three dice which give a total score of 5 , and hence show that the probability of Ivan obtaining a total score of 5 is \(\frac { 1 } { 36 }\).
2. Find the probability of Ivan obtaining a total score of 7.

4 Two fair dice are thrown.

1. Event \(A\) is 'the scores differ by 3 or more'. Find the probability of event \(A\).
2. Event \(B\) is 'the product of the scores is greater than 8 '. Find the probability of event \(B\).
3. State with a reason whether events \(A\) and \(B\) are mutually exclusive.

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.