2.03a Mutually exclusive and independent events

1 For the events \(A\) and \(B , \mathrm { P } ( A ) = 0.3 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = c\), where \(c \neq 0\).

1. Find \(\mathrm { P } ( A \cap B )\) in terms of \(c\).
2. Find \(\mathrm { P } ( B \mid A )\) and deduce that \(0.1 \leqslant c \leqslant 0.4\).

1 For the mutually exclusive events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = x\), where \(x \neq 0\).

1. Show that \(x \leqslant \frac { 1 } { 2 }\).
2. Show that \(A\) and \(B\) are not independent. The event \(C\) is independent of \(A\) and also independent of \(B\), and \(\mathrm { P } ( C ) = 2 x\).
3. Show that \(\mathrm { P } ( A \cup B \cup C ) = 4 x ( 1 - x )\).

3 For the events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = \frac { 3 } { 4 }\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = \frac { 1 } { 2 }\).

1. Find \(\mathrm { P } ( A \cap B )\). For a third event \(C , \mathrm { P } ( C ) = \frac { 1 } { 4 }\) and \(C\) is independent of the event \(A \cap B\).
2. Find \(\mathrm { P } ( A \cap B \cap C )\).
3. Given that \(\mathrm { P } ( C \mid A ) = \lambda\) and \(\mathrm { P } ( B \mid C ) = 3 \lambda\), and that no event occurs outside \(A \cup B \cup C\), find the value of \(\lambda\).

5. A group of 100 students are asked if they like folk music, rock music or soul music. \begin{displayquote} All students who like folk music also like rock music No students like both rock music and soul music 75 students do not like soul music 12 students who like rock music do not like folk music 30 students like folk music

1. Draw a Venn diagram to illustrate this information.
2. State two of these types of music that are mutually exclusive. \end{displayquote} Find the probability that a randomly chosen student
3. does not like folk music, rock music or soul music,
4. likes rock music,
5. likes folk music or soul music. Given that a randomly chosen student likes rock music,
6. find the probability that he or she also likes folk music.

1. The discrete random variable \(X\) has probability function \(\mathrm { p } ( x )\) and cumulative distribution function \(\mathrm { F } ( x )\) given in the table below.

1. Write down the value of \(d\)
2. Find the values of \(a\), \(b\) and \(c\)
3. Write down the value of \(\mathrm { P } ( X > 4 )\) Two independent observations, \(X _ { 1 }\) and \(X _ { 2 }\), are taken from the distribution of \(X\).
4. Find the probability that \(X _ { 1 }\) and \(X _ { 2 }\) are both odd. Given that \(X _ { 1 }\) and \(X _ { 2 }\) are both odd,
5. find the probability that the sum of \(X _ { 1 }\) and \(X _ { 2 }\) is 6 Give your answer to 3 significant figures.

4. Events \(A\) and \(B\) are shown in the Venn diagram below
where \(x , y , 0.10\) and 0.32 are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{c58f3e88-2dbc-40d6-a966-a5765a7c67ba-08_467_798_408_575}

1. Find an expression in terms of \(x\) for
   1. \(\mathrm { P } ( A )\)
   2. \(\mathrm { P } ( B \mid A )\)
2. Find an expression in terms of \(x\) and \(y\) for \(\mathrm { P } ( A \cup B )\) Given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\)
3. find the value of \(x\) and the value of \(y\)

2. \begin{figure}[h] \end{figure} Figure 1 shows part of a box and whisker plot for the marks in an examination with a large number of candidates. Part of the lower whisker has been torn off.

1. Given that \(75 \%\) of the candidates passed the examination, state the lowest mark for the award of a pass.
2. Given that the top \(25 \%\) of the candidates achieved a merit grade, state the lowest mark for the award of a merit grade. An outlier is defined as any value greater than \(c\) or any value less than \(d\) where $$\begin{aligned} & c = Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \\ & d = Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \end{aligned}$$
3. Find the value of \(c\) and the value of \(d\).
4. Write down the 3 highest marks scored in the examination. The 3 lowest marks in the examination were 5, 10 and 15
5. On the diagram on page 7, complete the box and whisker plot. Three candidates are selected at random from those who took this examination.
6. Find the probability that all 3 of these candidates passed the examination but only 2 achieved a merit grade.
   \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-05_285_1628_2343_166} Turn over for a spare diagram if you need to redraw your plot.

2. (a) Shade the region representing the event \(A \cup B ^ { \prime }\) on the Venn diagram below. \includegraphics[max width=\textwidth, alt={}, center]{01259350-0119-4500-a81b-bfa1b4234559-06_355_563_306_694} The two events \(C\) and \(D\) are mutually exclusive.
Given that \(\mathrm { P } ( C ) = \frac { 1 } { 5 }\) and \(\mathrm { P } ( D ) = \frac { 3 } { 10 }\) find
(b) (i) \(\quad \mathrm { P } ( C \cup D )\) (ii) \(\mathrm { P } ( C \mid D )\) The two events \(F\) and \(G\) are independent.
Given that \(\mathrm { P } ( F ) = \frac { 1 } { 6 }\) and \(\mathrm { P } ( F \cup G ) = \frac { 3 } { 8 }\) find
(c) (i) \(\mathrm { P } ( G )\) (ii) \(\mathrm { P } \left( F \mid G ^ { \prime } \right)\)

1. The Venn diagram shows the events \(A , B\) and \(C\) and their associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{4f034b9a-94c8-42f2-bd77-9adec277aba6-02_584_1061_296_445}

Find

1. \(\mathrm { P } \left( B ^ { \prime } \right)\)
2. \(\mathrm { P } ( A \cup C )\)
3. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\)

1. 1. In the Venn diagram below, \(A\) and \(B\) represent events and \(p , q , r\) and \(s\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{c316fa29-dedc-4890-bd82-31eb0bb819f9-12_400_789_347_639}

$$\mathrm { P } ( A ) = \frac { 7 } { 25 } \quad \mathrm { P } ( B ) = \frac { 1 } { 5 } \quad \mathrm { P } \left[ \left( A \cap B ^ { \prime } \right) \cup \left( A ^ { \prime } \cap B \right) \right] = \frac { 8 } { 25 }$$

1. Use algebra to show that \(2 p + 2 q + 2 r = \frac { 4 } { 5 }\)
2. Find the value of \(p\), the value of \(q\), the value of \(r\) and the value of \(s\) (ii) Two events, \(C\) and \(D\), are such that $$\mathrm { P } ( C ) = \frac { x } { x + 5 } \quad \mathrm { P } ( D ) = \frac { 5 } { x }$$ where \(x\) is a positive constant.
   By considering \(\mathrm { P } ( C ) + \mathrm { P } ( D )\) show that \(C\) and \(D\) cannot be mutually exclusive.

1. The events \(A\) and \(B\) satisfy

$$\mathrm { P } ( A ) = x \quad \mathrm { P } ( B ) = y \quad \mathrm { P } ( A \cup B ) = 0.65 \quad \mathrm { P } ( B \mid A ) = 0.3$$

1. Show that $$14 x + 20 y = 13$$ The events \(B\) and \(C\) are mutually exclusive such that $$\mathrm { P } ( B \cup C ) = 0.85 \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } x + y$$
2. 1. Find a second equation in \(x\) and \(y\)
   2. Hence find the value of \(x\) and the value of \(y\)
3. Determine whether or not \(A\) and \(B\) are statistically independent. You must show your working clearly.

6. The Venn diagram below shows the probabilities of customers having various combinations of a starter, main course or dessert at Polly's restaurant. \(S =\) the event a customer has a starter. \(M =\) the event a customer has a main course. \(D =\) the event a customer has a dessert. \includegraphics[max width=\textwidth, alt={}, center]{fa0dbe16-ace8-4c44-8404-2bc4e1879d57-10_602_1125_607_413} Given that the events \(S\) and \(D\) are statistically independent

1. find the value of \(p\).
2. Hence find the value of \(q\).
3. Find
   1. \(\quad\) P( \(D \mid M \cap S\) )
   2. \(\operatorname { P } \left( D \mid M \cap S ^ { \prime } \right)\) One evening 63 customers are booked into Polly's restaurant for an office party. Polly has asked for their starter and main course orders before they arrive. Of these 63 customers 27 ordered a main course and a starter, 36 ordered a main course without a starter.
4. Estimate the number of desserts that these 63 customers will have.

1. A company employs 90 administrators. The length of time that they have been employed by the company and their gender are summarised in the table below.

One of the 90 administrators is selected at random.

1. Find the probability that the administrator is female.
2. Given that the administrator has been employed by the company for less than 4 years, find the probability that this administrator is male.
3. Given that the administrator has been employed by the company for less than 10 years, find the probability that this administrator is male.
4. State, with a reason, whether or not the event 'selecting a male' is independent of the event 'selecting an administrator who has been employed by the company for less than 4 years'.

1. A bag contains 19 red beads and 1 blue bead only.

Linda selects a bead at random from the bag. She notes its colour and replaces the bead in the bag. She then selects a second bead at random from the bag and notes its colour. Find the probability that

1. both beads selected are blue,
2. exactly one bead selected is red. In another bag there are 9 beads, 4 of which are green and the rest are yellow.
   Linda selects 3 beads from this bag at random without replacement.
3. Find the probability that 2 of these beads are yellow and 1 is green. Linda replaces the 3 beads and then selects another 4 at random without replacement.
4. Find the probability that at least 1 of the beads is green.

4. \(\quad\) The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 2 } { 5 } , \mathrm { P } ( B ) = \frac { 1 } { 2 }\) and \(\mathrm { P } \left( A \quad B ^ { \prime } \right) = \frac { 4 } { 5 }\).

1. Find
   1. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
   2. \(\mathrm { P } ( A \cap B )\),
   3. \(\mathrm { P } ( A \cup B )\),
   4. \(\mathrm { P } \left( \begin{array} { l l } A & B \end{array} \right)\).
2. State, with a reason, whether or \(\operatorname { not } A\) and \(B\) are
   1. mutually exclusive,
   2. independent.

5. \begin{figure}[h] \end{figure} Figure 2 shows a partially completed Venn diagram of sports that a year group of students enjoy,where \(a , b , c , d\) and \(e\) are non-negative integers. The diagram shows how many students enjoy a combination of football( \(F\) ),golf( \(G\) ) and hockey \(( H )\) or none of these sports. There are \(n\) students in the year group.
It is known that
- \(\mathrm { P } ( F ) = \frac { 3 } { 7 }\) - \(\mathrm { P } ( H \mid G ) = \frac { 1 } { 3 }\) -\(F\) is independent of \(H \cap G\)

1. Show that \(\mathrm { P } ( F \cap H \cap G ) = \frac { 1 } { 7 } \mathrm { P } ( G )\)
2. Prove that if two events \(X\) and \(Y\) are independent,then \(X ^ { \prime }\) and \(Y\) are also independent.
3. Hence find the value \(k\) such that \(\mathrm { P } \left( F ^ { \prime } \cap H \cap G \right) = k \mathrm { P } ( G )\)
4. Show that \(c = \frac { 4 } { 3 } a\) Given further that \(\mathrm { P } ( F \mid H ) = \frac { 1 } { 5 }\)
5. find an expression for \(d\) in terms of \(a\) ,and hence deduce the maximum possible value of \(a\) .
6. Determine the possible values of \(n\) .

3 The table shows the joint probability distribution of two random variables \(X\) and \(Y\).

1. Find \(\operatorname { Cov } ( X , Y )\).
2. Are \(X\) and \(Y\) independent? Give a reason for your answer.
3. Find \(\mathrm { P } ( X = 1 \mid X Y = 2 )\).

5 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = 0.75\).

1. Find \(\mathrm { P } ( A \cap B )\) and \(\mathrm { P } ( A \cup B )\).
2. Determine, giving a reason in each case,
   1. whether \(A\) and \(B\) are mutually exclusive,
   2. whether \(A\) and \(B\) are independent.
   3. A further event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1\) and \(\mathrm { P } ( A \cap B \cap C ) = 0.05\). It is also given that \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C \right) = \mathrm { P } \left( A ^ { \prime } \cap B \cap C \right) = x\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 2 x\).
      Find \(\mathrm { P } ( C )\).

3 For events \(A , B\) and \(C\) it is given that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.5 , \mathrm { P } ( C ) = 0.4\) and \(\mathrm { P } ( A \cap B \cap C ) = 0.1\). It is also given that events \(A\) and \(B\) are independent and that events \(A\) and \(C\) are independent.

1. Find \(\mathrm { P } ( B \mid A )\).
2. Given also that events \(B\) and \(C\) are independent, find \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \cap C ^ { \prime } \right)\).
3. Given instead that events \(B\) and \(C\) are not independent, find the greatest and least possible values of \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \cap C ^ { \prime } \right)\).

8 A game uses an unbiased die with faces numbered 1 to 6 . The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.

1. Find the probability that the final score is 4 .
2. Given that the die is thrown only once, find the probability that the final score is 4 .
3. Given that the die is thrown twice, find the probability that the final score is 4 .

4 Jenny and Omar are each allowed two attempts at a high jump.

1. The probability that Jenny will succeed on her first attempt is 0.6 . If she fails on her first attempt, the probability that she will succeed on her second attempt is 0.7 . Calculate the probability that Jenny will succeed.
2. The probability that Omar will succeed on his first attempt is \(p\). If he fails on his first attempt, the probability that he will succeed on his second attempt is also \(p\). The probability that he succeeds is 0.51 . Find \(p\). \(530 \%\) of packets of Natural Crunch Crisps contain a free gift. Jan buys 5 packets each week.

8 Ann, Bill, Chris and Dipak play a game with a fair cubical die. Starting with Ann they take turns, in alphabetical order, to throw the die. This process is repeated as many times as necessary until a player throws a 6 . When this happens, the game stops and this player is the winner. Find the probability that

1. Chris wins on his first throw,
2. Dipak wins on his second throw,
3. Ann gets a third throw,
4. Bill throws the die exactly three times.