2.03c Conditional probability: using diagrams/tables

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.

1 Juan goes to college each day by any one of car or bus or walking. The probability that he goes by car is 0.2 , the probability that he goes by bus is 0.45 and the probability that he walks is 0.35 . When Juan goes by car, the probability that he arrives early is 0.6 . When he goes by bus, the probability that he arrives early is 0.1 . When he walks he always arrives early.

1. Draw a fully labelled tree diagram to represent this information.
2. Find the probability that Juan goes to college by car given that he arrives early.

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.

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 \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).

2 Seva has a coin which is biased so that when it is thrown the probability of obtaining a head is \(\frac { 1 } { 3 }\). He also has a bag containing 4 red marbles and 5 blue marbles. Seva throws the coin. If he obtains a head, he selects one marble from the bag at random. If he obtains a tail, he selects two marbles from the bag at random and without replacement.

1. Find the probability that Seva selects at least one red marble.
2. Find the probability that Seva obtains a head given that he selects no red marbles.

3 Box \(A\) contains 6 green balls and 3 yellow balls.
Box \(B\) contains 4 green 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 \(B\).

1. Draw a tree diagram to represent this information, showing the probability on each of the branches.
   [0pt] [4] \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-06_2727_38_132_2010}
   The probability that both the balls chosen are the same colour is \(\frac { 8 } { 15 }\).
2. Find the value of \(x\).

6 Box \(A\) contains 7 red balls and 1 blue ball. Box \(B\) contains 9 red balls and 5 blue 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.

1. Complete the tree diagram to show the probabilities. Box \(A\) \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-08_624_428_667_621} \section*{Box \(B\)} Red Blue Red Blue
2. Find the probability that the two balls chosen are not the same colour.
3. Find the probability that the ball chosen from box \(A\) is blue given that the ball chosen from box \(B\) is blue.

2 Georgie has a red scarf, a blue scarf and a yellow scarf. Each day she wears exactly one of these scarves. The probabilities for the three colours are \(0.2,0.45\) and 0.35 respectively. When she wears a red scarf, she always wears a hat. When she wears a blue scarf, she wears a hat with probability 0.4 . When she wears a yellow scarf, she wears a hat with probability 0.3 .

1. Find the probability that on a randomly chosen day Georgie wears a hat.
2. Find the probability that on a randomly chosen day Georgie wears a yellow scarf given that she does not wear a hat.

1 A bag contains 9 blue marbles and 3 red marbles. One marble is chosen at random from the bag. If this marble is blue, it is replaced back into the bag. If this marble is red, it is not returned to the bag. A second marble is now chosen at random from the bag.

1. Find the probability that both the marbles chosen are red.
2. Find the probability that the first marble chosen is blue given that the second marble chosen is red.

2 The probability that a student at a large music college plays in the band is 0.6. For a student who plays in the band, the probability that she also sings in the choir is 0.3 . For a student who does not play in the band, the probability that she sings in the choir is \(x\). The probability that a randomly chosen student from the college does not sing in the choir is 0.58 .

1. Find the value of \(x\).
   Two students from the college are chosen at random.
2. Find the probability that both students play in the band and both sing in the choir.

4 The random variable \(X\) takes each of the values \(1,2,3,4\) with probability \(\frac { 1 } { 4 }\). Two independent values of \(X\) are chosen at random. If the two values of \(X\) are the same, the random variable \(Y\) takes that value. Otherwise, the value of \(Y\) is the larger value of \(X\) minus the smaller value of \(X\).

1. Draw up the probability distribution table for \(Y\).
2. Find the probability that \(Y = 2\) given that \(Y\) is even.

4 In a certain country, the weather each day is classified as fine or rainy. The probability that a fine day is followed by a fine day is 0.75 and the probability that a rainy day is followed by a fine day is 0.4 . The probability that it is fine on 1 April is 0.8 . The tree diagram below shows the possibilities for the weather on 1 April and 2 April.

1. Complete the tree diagram to show the probabilities. 1 April \includegraphics[max width=\textwidth, alt={}, center]{33c0bd01-f27b-424c-a78a-6f36178bc08c-08_601_405_706_408} 2 April Fine Rainy Fine Rainy
2. Find the probability that 2 April is fine.
   Let \(X\) be the event that 1 April is fine and \(Y\) be the event that 3 April is rainy.
3. Find the value of \(\mathrm { P } ( X \cap Y )\).
4. Find the probability that 1 April is fine given that 3 April is rainy.

5 The 8 letters in the word RESERVED are arranged in a random order.

1. Find the probability that the arrangement has V as the first letter and E as the last letter.
2. Find the probability that the arrangement has both Rs together given that all three Es are together.

3 For her bedtime drink, Suki has either chocolate, tea or milk with probabilities \(0.45,0.35\) and 0.2 respectively. When she has chocolate, the probability that she has a biscuit is 0.3 When she has tea, the probability that she has a biscuit is 0.6 . When she has milk, she never has a biscuit. Find the probability that Suki has tea given that she does not have a biscuit.

7 Box \(A\) contains 6 red balls and 4 blue balls. Box \(B\) contains \(x\) red balls and 9 blue balls. A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\).

1. Complete the tree diagram below, giving the remaining four probabilities in terms of \(x\). \includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-12_688_759_484_731}
2. Show that the probability that both balls chosen are blue is \(\frac { 4 } { x + 10 }\).
   It is given that the probability that both balls chosen are blue is \(\frac { 1 } { 6 }\).
3. Find the probability, correct to 3 significant figures, that the ball chosen from box \(A\) is red given that the ball chosen from box \(B\) is red.
   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 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 \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).

1 On any day, Kino travels to school by bus, by car or on foot with probabilities 0.2, 0.1 and 0.7 respectively. The probability that he is late when he travels by bus is \(x\). The probability that he is late when he travels by car is \(2 x\) and the probability that he is late when he travels on foot is 0.25 . The probability that, on a randomly chosen day, Kino is late is 0.235 .

1. Find the value of \(x\).
2. Find the probability that, on a randomly chosen day, Kino travels to school by car given that he is not late.

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.

6 Freddie has two bags of marbles.
Bag \(X\) contains 7 red marbles and 3 blue marbles.
Bag \(Y\) contains 4 red marbles and 1 blue marble.
Freddie chooses one of the bags at random. A marble is removed at random from that bag and not replaced. A new red marble is now added to each bag. A second marble is then removed at random from the same bag that the first marble had been removed from.

1. Draw a tree diagram to represent this information, showing the probability on each of the branches.
2. Find the probability that both of the marbles removed from the bag are the same colour.
3. Find the probability that bag \(Y\) is chosen given that the marbles removed are not both the same colour.

3 Tim has two bags of marbles, \(A\) and \(B\).
Bag \(A\) contains 8 white, 4 red and 3 yellow marbles.
Bag \(B\) contains 6 white, 7 red and 2 yellow marbles.
Tim also has an ordinary fair 6 -sided dice. He rolls the dice. If he obtains a 1 or 2 , he chooses two marbles at random from bag \(A\), without replacement. If he obtains a \(3,4,5\) or 6 , he chooses two marbles at random from bag \(B\), without replacement.

1. Find the probability that both marbles are white.
2. Find the probability that the two marbles come from bag \(B\) given that one is white and one is red. [4]

4 Rahul has two bags, \(X\) and \(Y\). Bag \(X\) contains 4 red marbles and 2 blue marbles. Bag \(Y\) contains 3 red marbles and 4 blue marbles. Rahul also has a coin which is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\). Rahul throws the coin.

• If he obtains a head, he chooses at random a marble from bag \(X\). He notes the colour and replaces the marble in bag \(X\). He then chooses at random a second marble from bag \(X\).
• If he obtains a tail, he chooses at random a marble from bag \(Y\). He notes the colour and discards the marble. He then chooses at random a second marble from bag \(Y\).
  1. Find the probability that the two marbles that Rahul chooses are the same colour. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-06_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-07_2725_35_99_20}
  2. Find the probability that the two marbles that Rahul chooses are both from bag \(Y\) given that both marbles are blue.
     

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.

5 On Mondays, Rani cooks her evening meal. She has a pizza, a burger or a curry with probabilities \(0.35,0.44,0.21\) respectively. When she cooks a pizza, Rani has some fruit with probability 0.3 . When she cooks a burger, she has some fruit with probability 0.8 . When she cooks a curry, she never has any fruit.

1. Draw a fully labelled tree diagram to represent this information.
2. Find the probability that Rani has some fruit.
3. Find the probability that Rani does not have a burger given that she does not have any fruit.