2.03b Probability diagrams: tree, Venn, sample space

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.

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.

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.

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.

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 game for two players is played using a fair 4-sided dice with sides numbered 1, 2, 3 and 4. One turn consists of throwing the dice repeatedly up to a maximum of three times. When a 4 is obtained, no further throws are made during that turn. A player who obtains a 4 in their turn scores 1 point.

1. Show that the probability that a player obtains a 4 in one turn is \(\frac { 37 } { 64 }\).
   Xeno and Yao play this game.
2. Find the probability that neither Xeno nor Yao score any points in their first two turns.
3. Xeno and Yao each have three turns. Find the probability that Xeno scores 2 more points than Yao. \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1548_376_349} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_67_1566_466_328} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_72_1570_735_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_72_1570_826_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_77_1570_913_324} ........................................................................................................................................ . ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_1187_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_67_1570_1279_324} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_62_1570_1462_324} .......................................................................................................................................... ......................................................................................................................................... . .......................................................................................................................................... .......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_71_1570_1905_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_74_1570_1994_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_76_1570_2083_324} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_74_1570_2359_324} ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_2542_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_2631_324}

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 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.

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.

6 Three coins \(A , B\) and \(C\) are each thrown once.

• Coins \(A\) and \(B\) are each biased so that the probability of obtaining a head is \(\frac { 2 } { 3 }\).
• Coin \(C\) is biased so that the probability of obtaining a head is \(\frac { 4 } { 5 }\).
  1. Show that the probability of obtaining exactly 2 heads and 1 tail is \(\frac { 4 } { 9 }\).
     

The random variable \(X\) is the number of heads obtained when the three coins are thrown.

Draw up the probability distribution table for \(X\).

Given that \(\mathrm { E } ( X ) = \frac { 32 } { 15 }\), find \(\operatorname { Var } ( X )\).

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

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.

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.
     

2 A box contains 10 pens of which 3 are new. A random sample of two pens is taken.

1. Show that the probability of getting exactly one new pen in the sample is \(\frac { 7 } { 15 }\).
2. Construct a probability distribution table for the number of new pens in the sample.
3. Calculate the expected number of new pens in the sample.

6 The people living in 3 houses are classified as children ( \(C\) ), parents ( \(P\) ) or grandparents ( \(G\) ). The numbers living in each house are shown in the table below.

1. All the people in all 3 houses meet for a party. One person at the party is chosen at random. Calculate the probability of choosing a grandparent.
2. A house is chosen at random. Then a person in that house is chosen at random. Using a tree diagram, or otherwise, calculate the probability that the person chosen is a grandparent.
3. Given that the person chosen by the method in part (ii) is a grandparent, calculate the probability that there is also a parent living in the house.

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.