2.03c Conditional probability: using diagrams/tables

2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability \(p\) of hitting the target. Successive shots are independent.

1. Determine the probability that Jeff wins the game
   i) with his first shot,
   ii) with his second shot.
2. Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
3. Find the range of values of \(p\) for which Mary is more likely to win the game than Jeff.

1. A group of 200 adults were asked whether they read cooking magazines, travel magazines or sport magazines.
   Their replies showed that

• 29 read only cooking magazines
• 33 read only travel magazines
• 42 read only sport magazines
• 17 read cooking magazines and sport magazines but not travel magazines
• 11 read travel magazines and sport magazines but not cooking magazines
• 22 read cooking magazines and travel magazines but not sport magazines
• 32 do not read cooking magazines, travel magazines or sport magazines
  1. Using this information, complete the Venn diagram on page 11

One of these adults was chosen at random.

Find the probability that this adult,

1. reads cooking magazines and travel magazines and sport magazines,
2. does not read cooking magazines. Given that this adult reads travel magazines,

find the probability that this adult also reads sport magazines.

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1. A box contains only red counters and black counters.

There are \(n\) red counters and \(n + 1\) black counters.
Two counters are selected at random, one at a time without replacement, from the box.

1. Complete the tree diagram for this information. Give your probabilities in terms of \(n\) where necessary. \includegraphics[max width=\textwidth, alt={}, center]{fe416f2e-bc81-444b-a0ca-f0eae9a8b149-24_940_1180_591_413}
2. Show that the probability that the two counters selected are different colours is $$\frac { n + 1 } { 2 n + 1 }$$ The probability that the two counters selected are different colours is \(\frac { 25 } { 49 }\)
3. Find the total number of counters in the box before any counters were selected. Given that the two counters selected are different colours,
4. find the probability that the 1st counter is black. You must show your working.

12 Two players, \(A\) and \(B\), are taking turns to shoot at a basket with a basketball. The winner of this game is the first player to score a basket. The probability that \(A\) scores a basket with any shot is \(\frac { 1 } { 4 }\) and the probability that \(B\) scores a basket with any shot is \(\frac { 1 } { 5 }\). Each shot is independent of all other shots. \(A\) shoots first.

1. Find
   1. the probability that \(B\) wins with his first shot,
   2. the probability that \(A\) wins with his second shot,
   3. the probability that \(A\) wins the game.
   4. \(R\) is the total number of shots taken by \(A\) and \(B\) up to and including the shot that scores a basket.
      (a) Show that the probability generating function of \(R\) is given by $$\mathrm { G } ( t ) = \frac { 5 t + 3 t ^ { 2 } } { 4 \left( 5 - 3 t ^ { 2 } \right) }$$ (b) Hence find \(\mathrm { E } ( R )\).

5 In an archery competition, competitors are allowed up to three attempts to hit the bulls-eye. No one who succeeds may try again. \(45 \%\) of those entering the competition hit the bulls-eye first time. For those who fail to hit it the first time, \(60 \%\) of those attempting it for the second time succeed in hitting it. For those who fail twice, only \(15 \%\) of those attempting it for the third time succeed in hitting it. By drawing a tree diagram, or otherwise,

1. find the probability that a randomly chosen competitor fails at all three attempts,
2. find the probability that a randomly chosen competitor fails at the first attempt but succeeds at either the second or third attempt,
3. find the probability that a randomly chosen competitor succeeds in hitting the bulls-eye,
4. find the probability that a randomly chosen competitor requires exactly two attempts given that the competitor is successful.

2 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }\) and \(\mathrm { P } ( B \mid A ) = \frac { 1 } { 4 }\).
Find

1. \(\mathrm { P } ( A \cap B )\),
2. \(\mathrm { P } ( B )\).

2

1. A music club has 200 members. 75 members play the piano, 130 members like Elgar, and 30 members do not play the piano, nor do they like Elgar.
   1. Calculate the probability that a member chosen at random plays the piano but does not like Elgar.
   2. Calculate the probability that a member chosen at random plays the piano given that this member likes Elgar.
2. The music club is organising a concert. The programme is to consist of 7 pieces of music which are to be selected from 9 classical pieces and 6 modern pieces. Find the number of different concert programmes than can be produced if
   1. there are no restrictions,
   2. the programme must consist of 5 classical pieces and 2 modern pieces,
   3. there are to be more modern pieces than classical pieces.

15 In order to be accepted on a university course, a student needs to pass three exams.
The probability that the student passes the first exam is \(\frac { 3 } { 4 }\).
For each of the second and third exams, the probability of passing the exam is

• the same as the probability of passing the preceding exam if the student passed the preceding exam,
• half of the probability of passing the preceding exam if the student failed the preceding exam.
  1. Draw a tree diagram to represent the above information.
  2. Find the probability that the student passes all three exams.
  3. Find the probability that the student passes at least two of the exams.
  4. Find the probability that the student passes the third exam given that exactly two of the three exams are passed.

The probability that it will rain on any given day is \(x\). If it is raining, the probability that Aran wears a hat is 0.8 and if it is not raining, the probability that he wears a hat is 0.3. Whether it is raining or not, if Aran wears a hat, the probability that he wears a scarf is 0.4. If he does not wear a hat, the probability that he wears a scarf is 0.1. The probability that on a randomly chosen day it is not raining and Aran is not wearing a hat or a scarf is 0.36. Find the value of \(x\). [3]

A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable \(X\) represents the number of geese chosen.

1. Draw up the probability distribution of \(X\). [3]
2. Show that \(\mathrm{E}(X) = \frac{8}{7}\) and calculate \(\mathrm{Var}(X)\). [3]
3. When the farmer's dog is let loose, it chases either the ducks with probability \(\frac{3}{5}\) or the geese with probability \(\frac{2}{5}\). If the dog chases the ducks there is a probability of \(\frac{1}{10}\) that they will attack the dog. If the dog chases the geese there is a probability of \(\frac{1}{4}\) that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese. [4]

Nikita goes shopping to buy a birthday present for her mother. She buys either a scarf, with probability 0.3, or a handbag. The probability that her mother will like the choice of scarf is 0.72. The probability that her mother will like the choice of handbag is \(x\). This information is shown on the tree diagram. The probability that Nikita's mother likes the present that Nikita buys is 0.783.

1. Find \(x\). [3]
2. Given that Nikita's mother does not like her present, find the probability that the present is a scarf. [4]

A bag contains a large number of coloured counters. Each counter is labelled A, B or C 30% of the counters are labelled A 45% of the counters are labelled B The rest of the counters are labelled C It is known that 2% of the counters labelled A are red 4% of the counters labelled B are red 6% of the counters labelled C are red One counter is selected at random from the bag.

1. Complete the tree diagram on the opposite page to illustrate this information. [2]
2. Calculate the probability that the counter is labelled A and is not red. [2]
3. Calculate the probability that the counter is red. [2]
4. Given that the counter is red, find the probability that it is labelled C [3]

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Jake and Kamil are sometimes late for school. The events \(J\) and \(K\) are defined as follows \(J =\) the event that Jake is late for school \(K =\) the event that Kamil is late for school \(\text{P}(J) = 0.25\), \(\text{P}(J \cap K) = 0.15\) and \(\text{P}(J' \cap K') = 0.7\) On a randomly selected day, find the probability that

1. at least one of Jake or Kamil are late for school, [1]
2. Kamil is late for school. [2]

Given that Jake is late for school,

1. find the probability that Kamil is late. [3]

The teacher suspects that Jake being late for school and Kamil being late for school are linked in some way.

1. Determine whether or not \(J\) and \(K\) are statistically independent. [2]
2. Comment on the teacher's suspicion in the light of your calculation in (d). [1]

For any married couple who are members of a tennis club, the probability that the husband has a degree is \(\frac{3}{5}\) and the probability that the wife has a degree is \(\frac{1}{2}\). The probability that the husband has a degree, given that the wife has a degree, is \(\frac{11}{12}\). A married couple is chosen at random.

1. Show that the probability that both of them have degrees is \(\frac{11}{24}\). [2]
2. Draw a Venn diagram to represent these data. [5]

Find the probability that

1. only one of them has a degree, [2]
2. neither of them has a degree. [3]

Two married couples are chosen at random.

1. Find the probability that only one of the two husbands and only one of the two wives have degrees. [6]

The values of the two variables \(A\) and \(B\) given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled \(A\) and \(B\). One card is selected from Pack \(A\). Let \(A_i\) represent the event that the first digit on this card is \(i\).

1. Write down the value of P\((A_2)\). [1 mark] The card taken from Pack \(A\) is now transferred into Pack \(B\), and one card is picked at random from Pack \(B\). Let \(B_i\) represent the event that the first digit on this card is \(i\).
2. Show that P\((A_1 \cap B_1) = \frac{1}{24}\). [3 marks]
3. Show that P\((A_6 | B_2) = \frac{4}{41}\). [5 marks]
4. Find the value of P\((A_1 \cup B_4)\). [5 marks]

The events \(A\) and \(B\) are such that P\((A \cap B) = 0.24\), P\((A \cup B) = 0.88\) and P\((B) = 0.52\).

1. Find P\((A)\). [3 marks]
2. Determine, with reasons, whether \(A\) and \(B\) are
   1. mutually exclusive,
   2. independent.
   [4 marks]
3. Find P\((B | A)\). [2 marks]
4. Find P\((A' | B')\). [3 marks]

Of the cars that are taken to a certain garage for an M.O.T. test, 87% pass. However, 2% of these have faults for which they should have been failed. 5% of the cars which fail are in fact roadworthy and should have passed. Using a tree diagram, or otherwise, calculate the probabilities that a car chosen at random

1. should have passed the test, regardless of whether it actually did or not, [4 marks]
2. failed the test, given that it should have passed. [3 marks]

The garage is told to improve its procedures. When it is inspected again a year later, it is found that the pass rate is still 87% overall and 2% of the cars passed have faults as before, but now 0.3% of the cars which should have passed are failed and \(x\)% of the cars which are failed should have passed.

1. Find the value of \(x\). [8 marks]

\(A\), \(B\) and \(C\) are three events such that \(\text{P}(A) = x\), \(\text{P}(B) = y\) and \(\text{P}(C) = x + y\). It is known that \(\text{P}(A \cup B) = 0.6\) and \(\text{P}(B \mid A) = 0.2\).

1. Show that \(4x + 5y = 3\). [2 marks]

It is also known that \(B\) and \(C\) are mutually exclusive and that \(\text{P}(B \cup C) = 0.9\)

1. Obtain another equation in \(x\) and \(y\) and hence find the values of \(x\) and \(y\). [4 marks]
2. Deduce whether or not \(A\) and \(B\) are independent events. [2 marks]

A darts player throws two darts, attempting to score a bull's-eye with each. The probability that he will achieve this with his first dart is \(0.25\). If he misses with his first dart, the probability that he will also miss with his second dart is \(0.7\). The probability that he will miss with at least one dart is \(0.9\).

1. Show that the probability that he succeeds with his first dart but misses with his second is \(0.15\). [5 marks]
2. Find the conditional probability that he misses with both darts, given that he misses with at least one. [3 marks]

Among the families with two children in a large city, the probability that the elder child is a boy is \(\frac{5}{12}\) and the probability that the younger child is a boy is \(\frac{9}{16}\). The probability that the younger child is a girl, given that the elder child is a girl, is \(\frac{1}{4}\). One of the families is chosen at random. Using a tree diagram, or otherwise,

1. show that the probability that both children are boys is \(\frac{1}{8}\). [5 marks]

Find the probability that

1. one child is a boy and the other is a girl, [3 marks]
2. one child is a boy given that the other is a girl. [3 marks]

If three of the families are chosen at random,

1. find the probability that exactly two of the families have two boys. [3 marks]
2. State an assumption that you have made in answering part (d). [1 mark]

The students in a large Sixth Form can choose to do exactly one of Community Service, Games or Private Study on Wednesday afternoons. The probabilities that a randomly chosen student does Games and Private Study are \(\frac{3}{8}\) and \(\frac{1}{5}\) respectively. It may be assumed that the number of students is large enough for these probabilities to be treated as constant.

1. Find the probability that a randomly chosen student does Community Service. [2 marks]
2. If two students are chosen at random, find the probability that they both do the same activity. [3 marks]
3. If three students are chosen at random, find the probability that exactly one of them does Games. [3 marks]

Two-fifths of the students are girls, and a quarter of these girls do Private Study.

1. Find the probability that a randomly chosen student who does Private Study is a boy. [5 marks]

The table shows the numbers of male and female members of a vintage car club who own either a Jaguar or a Bentley. No member owns both makes of car.

	Male	Female
Jaguar	25	15
Bentley	12	8

One member is chosen at random from these 60 members.

1. Given that this member is male, find the probability that he owns a Jaguar. [2]

Now two members are chosen at random from the 60 members. They are chosen one at a time, without replacement.

1. Given that the first one of these members is female, find the probability that both own Jaguars. [4]

In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random. • \(G\) is the event that this person goes to the gym. • \(R\) is the event that this person goes running. You are given that P(G) = 0.24, P(R) = 0.13 and P(G ∩ R) = 0.06.

1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(G\) and \(R\) are independent. [2]
3. Find P(R | G). [3]