2.03a Mutually exclusive and independent events

8 A random sample of 200 is taken from the adult population of a town and classified by age-group and preferred type of car. The results are given in the following table. Test, at the \(5 \%\) significance level, whether preferred type of car is independent of age-group.

10 Customers were asked which of three brands of coffee, \(A , B\) and \(C\), they prefer. For a random sample of 80 male customers and 60 female customers, the numbers preferring each brand are shown in the following table. Test, at the \(5 \%\) significance level, whether there is a difference between coffee preferences of male and female customers. A larger random sample is now taken. It consists of \(80 n\) male customers and \(60 n\) female customers, where \(n\) is a positive integer. It is found that the proportions choosing each brand are identical to those in the smaller sample. Find the least value of \(n\) that would lead to a different conclusion for the 5\% significance level hypothesis test.

2 A total of 500 students were asked which one of four colleges they attended and whether they preferred soccer or hockey. The numbers of students in each category are shown in the following table.

1. Find the probability that a randomly chosen student is at Canton college and prefers hockey.
2. Find the probability that a randomly chosen student is at Devar college given that he prefers soccer.
3. One of the students is chosen at random. Determine whether the events 'the student prefers hockey' and 'the student is at Amos college or Benn college' are independent, justifying your answer.

4 To gain a place at a science college, students first have to pass a written test and then a practical test.
Each student is allowed a maximum of two attempts at the written test. A student is only allowed a second attempt if they fail the first attempt. No student is allowed more than one attempt at the practical test. If a student fails both attempts at the written test, then they cannot attempt the practical test. The probability that a student will pass the written test at the first attempt is 0.8 . If a student fails the first attempt at the written test, the probability that they will pass at the second attempt is 0.6 . The probability that a student will pass the practical test is always 0.3 .

1. Draw a tree diagram to represent this information, showing the probabilities on the branches.
2. Find the probability that a randomly chosen student will succeed in gaining a place at the college.
   [0pt] [2]
3. Find the probability that a randomly chosen student passes the written test at the first attempt given that the student succeeds in gaining a place at the college.

3 On each day that Alexa goes to work, the probabilities that she travels by bus, by train or by car are \(0.4,0.35\) and 0.25 respectively. When she travels by bus, the probability that she arrives late is 0.55 . When she travels by train, the probability that she arrives late is 0.7 . When she travels by car, the probability that she arrives late is \(x\). On a randomly chosen day when Alexa goes to work, the probability that she does not arrive late is 0.48 .

1. Find the value of \(x\).
2. Find the probability that Alexa travels to work by train given that she arrives late.

7 In the region of Arka, the total number of households in the three villages Reeta, Shan and Teber is 800 . Each of the households was asked about the quality of their broadband service. Their responses are summarised in the following table.

1. 1. Find the probability that a randomly chosen household is in Shan and has poor broadband service.
   2. Find the probability that a randomly chosen household has good broadband service given that the household is in Shan.
      In the whole of Arka there are a large number of households. A survey showed that \(35 \%\) of households in Arka have no broadband service.
2. 1. 10 households in Arka are chosen at random. Find the probability that fewer than 3 of these households have no broadband service.
   2. 120 households in Arka are chosen at random. Use an approximation to find the probability that more than 32 of these households have no broadband service.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Janice is playing a computer game. She has to complete level 1 and level 2 to finish the game. She is allowed at most two attempts at any level.

• For level 1 , the probability that Janice completes it at the first attempt is 0.6 . If she fails at her first attempt, the probability that she completes it at the second attempt is 0.3 .
• If Janice completes level 1, she immediately moves on to level 2.
• For level 2, the probability that Janice completes it at the first attempt is 0.4 . If she fails at her first attempt, the probability that she completes it at the second attempt is 0.2 .
  1. Show that the probability that Janice moves on to level 2 is 0.72 .
  2. Find the probability that Janice finishes the game.
  3. Find the probability that Janice fails exactly one attempt, given that she finishes the game.
     

If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Sajid is practising for a long jump competition. He counts any jump that is longer than 6 m as a success. On any day, the probability that he has a success with his first jump is 0.2 . For any subsequent jump, the probability of a success is 0.3 if the previous jump was a success and 0.1 otherwise. Sajid makes three jumps.

1. Draw a tree diagram to illustrate this information, showing all the probabilities.
2. Find the probability that Sajid has exactly one success given that he has at least one success.
   On another day, Sajid makes six jumps.
3. Find the probability that only his first three jumps are successes or only his last three jumps are successes.

3

1. Find the number of different arrangements of the 8 letters in the word COCOONED.
2. Find the number of different arrangements of the 8 letters in the word COCOONED in which the first letter is O and the last letter is N .
3. Find the probability that a randomly chosen arrangement of the 8 letters in the word COCOONED has all three Os together given that the two Cs are next to each other.

2 A sports event is taking place for 4 days, beginning on Sunday. The probability that it will rain on Sunday is 0.4 . On any subsequent day, the probability that it will rain is 0.7 if it rained on the previous day and 0.2 if it did not rain on the previous day.

1. Find the probability that it does not rain on any of the 4 days of the event.
2. Find the probability that the first day on which it rains during the event is Tuesday.
3. Find the probability that it rains on exactly one of the 4 days of the event.

4 A fair 5 -sided spinner has sides labelled 1, 2, 3, 4, 5. The spinner is spun repeatedly until a 2 is obtained on the side on which the spinner lands. The random variable \(X\) denotes the number of spins required.

1. Find \(\mathrm { P } ( X = 4 )\).
2. Find \(\mathrm { P } ( X < 6 )\).
   Two fair 5 -sided spinners, each with sides labelled \(1,2,3,4,5\), are spun at the same time. If the numbers obtained are equal, the score is 0 . Otherwise, the score is the higher number minus the lower number.
3. Find the probability that the score is greater than 0 given that the score is not equal to 2 .
   The two spinners are spun at the same time repeatedly .
4. For 9 randomly chosen spins of the two spinners, find the probability that the score is greater than 2 on at least 3 occasions.

5 Jasmine throws two ordinary fair 6-sided dice at the same time and notes the numbers on the uppermost faces. The events \(A\) and \(B\) are defined as follows. \(A\) : The sum of the two numbers is less than 6 . \(B : \quad\) The difference between the two numbers is at most 2 .

1. Determine whether or not the events \(A\) and \(B\) are independent.
2. Find \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).

6

1. How many different arrangements are there of the 9 letters in the word RECORDERS?
2. How many different arrangements are there of the 9 letters in the word RECORDERS in which there is an E at the beginning, an E at the end and the three Rs are not all together? \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-12_2725_40_136_2007}
   The 9 letters of the word RECORDERS are divided at random into two groups: a group of 5 letters and a group of 4 letters.
3. Find the probability that the three Rs are in the same group.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-14_2715_35_143_2012}

7 There are 400 students at a school in a certain country. Each student was asked whether they preferred swimming, cycling or running and the results are given in the following table. A student is chosen at random.

1. 1. Find the probability that the student prefers swimming.
   2. Determine whether the events 'the student is male' and 'the student prefers swimming' are independent, justifying your answer.
      On average at all the schools in this country \(30 \%\) of the students do not like any sports.
2. 1. 10 of the students from this country are chosen at random. Find the probability that at least 3 of these students do not like any sports.
   2. 90 students from this country are now chosen at random. Use an approximation to find the probability that fewer than 32 of them do not like any sports.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A factory produces chocolates in three flavours: lemon, orange and strawberry in the ratio \(3 : 5 : 7\) respectively. Nell checks the chocolates on the production line by choosing chocolates randomly one at a time.

1. Find the probability that the first chocolate with lemon flavour that Nell chooses is the 7th chocolate that she checks.
2. Find the probability that the first chocolate with lemon flavour that Nell chooses is after she has checked at least 6 chocolates.
   'Surprise' boxes of chocolates each contain 15 chocolates: 3 are lemon, 5 are orange and 7 are strawberry. Petra has a box of Surprise chocolates. She chooses 3 chocolates at random from the box. She eats each chocolate before choosing the next one.
3. Find the probability that none of Petra's 3 chocolates has orange flavour.
4. Find the probability that each of Petra's 3 chocolates has a different flavour.
5. Find the probability that at least 2 of Petra's 3 chocolates have strawberry flavour given that none of them has orange flavour.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 Two ordinary fair dice, one red and the other blue, are thrown.
Event \(A\) is 'the score on the red die is divisible by 3 '.
Event \(B\) is 'the sum of the two scores is at least 9 '.

1. Find \(\mathrm { P } ( A \cap B )\).
2. Hence determine whether or not the events \(A\) and \(B\) are independent.

5 A game is played with an ordinary fair 6-sided die. A player throws the die once. If the result is \(2,3,4\) or 5 , that result is the player's score and the player does not throw the die again. If the result is 1 or 6 , the player throws the die a second time and the player's score is the sum of the two numbers from the two throws.

1. Draw a fully labelled tree diagram to represent this information. Events \(A\) and \(B\) are defined as follows. \(A\) : the player's score is \(5,6,7,8\) or 9 \(B\) : the player has two throws
2. Show that \(\mathrm { P } ( A ) = \frac { 1 } { 3 }\).
3. Determine whether or not events \(A\) and \(B\) are independent.
4. Find \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).

5 Eric has three coins. One of the coins is fair. The other two coins are each biased so that the probability of obtaining a head on any throw is \(\frac { 1 } { 4 }\), independently of all other throws. Eric throws all three coins at the same time. Events \(A\) and \(B\) are defined as follows. \(A\) : all three coins show the same result \(B\) : at least one of the biased coins shows a head

1. Show that \(\mathrm { P } ( B ) = \frac { 7 } { 16 }\).
2. Find \(\mathrm { P } ( A \mid B )\).
   The random variable \(X\) is the number of heads obtained when Eric throws the three coins.
3. Draw up the probability distribution table for \(X\).

7 Sam and Tom are playing a game which involves a bag containing 5 white discs and 3 red discs. They take turns to remove one disc from the bag at random. Discs that are removed are not replaced into the bag. The game ends as soon as one player has removed two red discs from the bag. That player wins the game. Sam removes the first disc.

1. Find the probability that Tom removes a red disc on his first turn.
2. Find the probability that Tom wins the game on his second turn.
3. Find the probability that Sam removes a red disc on his first turn given that Tom wins the game on his second turn.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A red spinner has four sides labelled \(1,2,3,4\). When the spinner is spun, the score is the number on the side on which it lands. The random variable \(X\) denotes this score. The probability distribution table for \(X\) is given below.

1. Show that \(p = 0.12\).
   A fair blue spinner and a fair green spinner each have four sides labelled 1, 2, 3, 4. All three spinners (red, blue and green) are spun at the same time.
2. Find the probability that the sum of the three scores is 4 or less.
3. Find the probability that the product of the three scores is 4 or less given that \(X\) is odd.

1 At a college, the students choose exactly one of tennis, hockey or netball to play. The table shows the numbers of students in Year 1 and Year 2 at the college playing each of these sports. One student is chosen at random from the 120 students. Events \(X\) and \(N\) are defined as follows: \(X\) : the student is in Year 1 \(N\) : the student plays netball.

1. Find \(\mathrm { P } ( X \mid N )\).
2. Find \(\mathrm { P } ( N \mid X )\).
3. Determine whether or not \(X\) and \(N\) are independent events.
   One of the students who plays netball takes 8 shots at goal. On each shot, the probability that she will succeed is 0.15 , independently of all other shots.
4. Find the probability that she succeeds on fewer than 3 of these shots.

1 The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.

1. Show that the probability that the score is 4 is \(\frac { 1 } { 12 }\).
   The two dice are thrown repeatedly until a score of 4 is obtained. The number of throws taken is denoted by the random variable \(X\).
2. Find the mean of \(X\).
3. Find the probability that a score of 4 is first obtained on the 6th throw.
4. Find \(\mathrm { P } ( X < 8 )\).

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.

7 A vegetable basket contains 12 peppers, of which 3 are red, 4 are green and 5 are yellow. Three peppers are taken, at random and without replacement, from the basket.

1. Find the probability that the three peppers are all different colours.
2. Show that the probability that exactly 2 of the peppers taken are green is \(\frac { 12 } { 55 }\).
3. The number of green peppers taken is denoted by the discrete random variable \(X\). Draw up a probability distribution table for \(X\).

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 \(B\).