2.03d Calculate conditional probability: from first principles

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.

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.

1 The numbers on the faces of a fair six-sided dice are \(1,2,2,3,3,3\). The random variable \(X\) is the total score when the dice is rolled twice.

1. Draw up the probability distribution table for \(X\).
2. Find the value of \(\operatorname { Var } ( X )\). \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-02_2714_34_143_2012}
3. Find the probability that \(X\) is even given that \(X > 3\).

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 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.

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.

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 A factory produces chocolates. 30\% of the chocolates are wrapped in gold foil, 25\% are wrapped in red foil and the remainder are unwrapped. Indigo chooses 8 chocolates at random from the production line.

1. Find the probability that she obtains no more than 2 chocolates that are wrapped in red foil.
   Jake chooses chocolates one at a time at random from the production line.
2. Find the probability that the first time he obtains a chocolate that is wrapped in red foil is before the 7th choice. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-08_2720_35_106_2015} \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-09_2717_29_105_22} Keifa chooses chocolates one at a time at random from the production line.
3. Find the probability that the second chocolate chosen is the first one wrapped in gold foil given that the fifth chocolate chosen is the first unwrapped chocolate.

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 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.

2 The probability that Henk goes swimming on any day is 0.2 . On a day when he goes swimming, the probability that Henk has burgers for supper is 0.75 . On a day when he does not go swimming the probability that he has burgers for supper is \(x\). This information is shown on the following tree diagram. \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-2_693_1038_845_555} The probability that Henk has burgers for supper on any day is 0.5 .

1. Find \(x\).
2. Given that Henk has burgers for supper, find the probability that he went swimming that day.

2 Jamie is equally likely to attend or not to attend a training session before a football match. If he attends, he is certain to be chosen for the team which plays in the match. If he does not attend, there is a probability of 0.6 that he is chosen for the team.

1. Find the probability that Jamie is chosen for the team.
2. Find the conditional probability that Jamie attended the training session, given that he was chosen for the team.

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 \(B\).