2.03d Calculate conditional probability: from first principles

5. A person's blood group is determined by whether or not it contains any of 3 substances \(A , B\) and \(C\). A doctor surveyed 300 patients' blood and produced the table below.

1. Draw a Venn diagram to represent this information.
2. Find the probability that a randomly chosen patient's blood contains substance \(C\). Harry is one of the patients. Given that his blood contains substance \(A\),
3. find the probability that his blood contains all 3 substances. Patients whose blood contains none of these substances are called universal blood donors.
4. Find the probability that a randomly chosen patient is a universal blood donor.

2. On a randomly chosen day the probability that Bill travels to school by car, by bicycle or on foot is \(\frac { 1 } { 2 } , \frac { 1 } { 6 }\) and \(\frac { 1 } { 3 }\) respectively. The probability of being late when using these methods of travel is \(\frac { 1 } { 5 } , \frac { 2 } { 5 }\) and \(\frac { 1 } { 10 }\) respectively.

1. Draw a tree diagram to represent this information.
2. Find the probability that on a randomly chosen day
   1. Bill travels by foot and is late,
   2. Bill is not late.
3. Given that Bill is late, find the probability that he did not travel on foot.

7. (a) Given that \(\mathrm { P } ( A ) = a\) and \(\mathrm { P } ( B ) = b\) express \(\mathrm { P } ( A \cup B )\) in terms of \(a\) and \(b\) when

1. \(A\) and \(B\) are mutually exclusive,
2. \(A\) and \(B\) are independent. Two events \(R\) and \(Q\) are such that \(\mathrm { P } \left( R \cap Q ^ { \prime } \right) = 0.15 , \quad \mathrm { P } ( Q ) = 0.35\) and \(\mathrm { P } ( R \mid Q ) = 0.1\) Find the value of
   (b) \(\mathrm { P } ( R \cup Q )\),
   (c) \(\mathrm { P } ( R \cap Q )\),
   (d) \(\mathrm { P } ( R )\).

2. An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, its colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability \(\frac { 2 } { 3 }\) of landing heads is spun.
When a blue ball is selected a fair coin is spun.

1. Complete the tree diagram below to show the possible outcomes and associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_787_395_734_548} \section*{Coin}
   \includegraphics[max width=\textwidth, alt={}]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-03_1007_488_808_950}
   Shivani selects a ball and spins the appropriate coin.
2. Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
3. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
4. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.

4. The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines \(A , B\) and \(C\). \begin{figure}[h] \end{figure} One of these students is selected at random.

1. Show that the probability that the student reads more than one magazine is \(\frac { 1 } { 6 }\).
2. Find the probability that the student reads \(A\) or \(B\) (or both).
3. Write down the probability that the student reads both \(A\) and \(C\). Given that the student reads at least one of the magazines,
4. find the probability that the student reads \(C\).
5. Determine whether or not reading magazine \(B\) and reading magazine \(C\) are statistically independent.

6. \begin{figure}[h] \end{figure} The Venn diagram in Figure 1 shows three events \(A , B\) and \(C\) and the probabilities associated with each region of \(B\). The constants \(p , q\) and \(r\) each represent probabilities associated with the three separate regions outside \(B\). The events \(A\) and \(B\) are independent.

1. Find the value of \(p\). Given that \(\mathrm { P } ( B \mid C ) = \frac { 5 } { 11 }\)
2. find the value of \(q\) and the value of \(r\).
3. Find \(\mathrm { P } ( A \cup C \mid B )\).

3. In a company the 200 employees are classified as full-time workers, part-time workers or contractors.
The table below shows the number of employees in each category and whether they walk to work or use some form of transport. The events \(F , H\) and \(C\) are that an employee is a full-time worker, part-time worker or contractor respectively. Let \(W\) be the event that an employee walks to work. An employee is selected at random.
Find

1. \(\mathrm { P } ( H )\)
2. \(\mathrm { P } \left( [ F \cap W ] ^ { \prime } \right)\)
3. \(\mathrm { P } ( W \mid C )\) Let \(B\) be the event that an employee uses the bus.
   Given that \(10 \%\) of full-time workers use the bus, \(30 \%\) of part-time workers use the bus and \(20 \%\) of contractors use the bus,
4. draw a Venn diagram to represent the events \(F , H , C\) and \(B\),
5. find the probability that a randomly selected employee uses the bus to travel to work.

1. \(\quad A\) and \(B\) are two events such that

$$\mathrm { P } ( B ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \mid B ) = \frac { 2 } { 5 } \quad \mathrm { P } ( A \cup B ) = \frac { 13 } { 20 }$$

1. Find \(\mathrm { P } ( A \cap B )\).
2. Draw a Venn diagram to show the events \(A , B\) and all the associated probabilities. Find
3. \(\mathrm { P } ( A )\)
4. \(\mathrm { P } ( B \mid A )\)
5. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\)

7. In a large company, 78\% of employees are car owners, \(30 \%\) of these car owners are also bike owners,
85\% of those who are not car owners are bike owners.

1. Draw a tree diagram to represent this information. An employee is selected at random.
2. Find the probability that the employee is a car owner or a bike owner but not both. Another employee is selected at random. Given that this employee is a bike owner,
3. find the probability that the employee is a car owner. Two employees are selected at random.
4. Find the probability that only one of them is a bike owner.

1. In a factory, three machines, \(J , K\) and \(L\), are used to make biscuits.

Machine \(J\) makes \(25 \%\) of the biscuits. Machine \(K\) makes \(45 \%\) of the biscuits. The rest of the biscuits are made by machine \(L\).
It is known that \(2 \%\) of the biscuits made by machine \(J\) are broken, \(3 \%\) of the biscuits made by machine \(K\) are broken and 5\% of the biscuits made by machine \(L\) are broken.

1. Draw a tree diagram to illustrate all the possible outcomes and associated probabilities. A biscuit is selected at random.
2. Calculate the probability that the biscuit is made by machine \(J\) and is not broken.
3. Calculate the probability that the biscuit is broken.
4. Given that the biscuit is broken, find the probability that it was not made by machine \(K\).
   

8. For the events \(A\) and \(B\), $$\mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.22 \text { and } \mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = 0.18$$

1. Find \(\mathrm { P } ( A )\).
2. Find \(\mathrm { P } ( A \cup B )\). Given that \(\mathrm { P } ( A \mid B ) = 0.6\)
3. find \(\mathrm { P } ( A \cap B )\).
4. Determine whether or not \(A\) and \(B\) are independent.

1. A college has 80 students in Year 12.

20 students study Biology
28 students study Chemistry
30 students study Physics
7 students study both Biology and Chemistry
11 students study both Chemistry and Physics
5 students study both Physics and Biology
3 students study all 3 of these subjects

1. Draw a Venn diagram to represent this information. A Year 12 student at the college is selected at random.
2. Find the probability that the student studies Chemistry but not Biology or Physics.
3. Find the probability that the student studies Chemistry or Physics or both. Given that the student studies Chemistry or Physics or both,
4. find the probability that the student does not study Biology.
5. Determine whether studying Biology and studying Chemistry are statistically independent.
   

4. The Venn diagram shows the probabilities of customer bookings at Harry's hotel. \(R\) is the event that a customer books a room \(B\) is the event that a customer books breakfast \(D\) is the event that a customer books dinner \(u\) and \(t\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{e3b92a5b-c0ad-4176-9b05-cb07a44aa265-08_604_1047_696_450}

1. Write down the probability that a customer books breakfast but does not book a room. Given that the events \(B\) and \(D\) are independent
2. find the value of \(t\)
3. hence find the value of \(u\)
4. Find
   1. \(\quad\) P( \(D \mid R \cap B\) )
   2. \(\mathrm { P } \left( D \mid R \cap B ^ { \prime } \right)\) A coach load of 77 customers arrive at Harry's hotel. Of these 77 customers 40 have booked a room and breakfast 37 have booked a room without breakfast
5. Estimate how many of these 77 customers will book dinner.

The Venn diagram shows three events \(A , B\) and \(C\), where \(p , q , r , s\) and \(t\) are probabilities. \includegraphics[max width=\textwidth, alt={}, center]{319667e7-3f8b-4a33-8fc5-ef72154d1421-10_647_972_306_488}
(b) Find the value of \(r\).
(c) Hence write down the value of \(s\) and the value of \(t\).
(d) State, giving a reason, whether or not the events \(A\) and \(B\) are independent.
(e) Find \(\mathrm { P } ( B \mid A \cup C )\). \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } ( C ) = 0.25\) and the events \(B\) and \(C\) are independent.
(a) Find the value of \(p\) and the value of \(q\).

4.A bag contains 64 coloured beads.There are \(r\) red beads,\(y\) yellow beads and 1 green bead and \(r + y + 1 = 64\) Two beads are selected at random,one at a time without replacement.

1. Find the probability that the green bead is one of the beads selected. The probability that both of the beads are red is \(\frac { 5 } { 84 }\)
2. Show that \(r\) satisfies the equation \(r ^ { 2 } - r - 240 = 0\)
3. Hence show that the only possible value of \(r\) is 16
4. Given that at least one of the beads is red,find the probability that they are both red.

5. A keep-fit enthusiast swims, runs or cycles each day with probabilities \(0.2,0.3\) and 0.5 respectively. If he swims he then spends time in the sauna with probability 0.35 . The probabilities that he spends time in the sauna after running or cycling are 0.2 and 0.45 respectively.

1. Represent this information on a tree diagram.
2. Find the probability that on any particular day he uses the sauna.
3. Given that he uses the sauna one day, find the probability that he had been swimming.
4. Given that he did not use the sauna one day, find the probability that he had been swimming.

7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \begin{cases} \frac { 1 } { 20 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ \frac { 1 } { 10 } ( 6 - x ) & 2 < x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(\mathrm { f } ( x )\) for all values of \(x\).
2. Write down the mode of \(X\).
3. Show that \(\mathrm { P } ( X > 2 ) = 0.8\)
4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\). Given that \(\mathrm { P } ( X < a \mid X > 2 ) = \frac { 5 } { 8 }\)
5. find the value of \(\mathrm { F } ( a )\).
6. Hence, or otherwise, find the value of \(a\). Give your answer to 3 significant figures.

1. A continuous random variable \(X\) has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 16 } ( x - 1 ) ^ { 2 } & 1 \leqslant x \leqslant 5 \\ 1 & x > 5 \end{array} \right.$$

1. Find \(\mathrm { P } ( X > 4 )\)
2. Find \(\mathrm { P } ( X > 3 \mid 2 < X < 4 )\)
3. Find the exact value of \(\mathrm { E } ( X )\)

1. The continuous random variable \(X\) has probability density function

$$f ( x ) = \begin{cases} c ( x + 3 ) & - 3 \leqslant x < 0 \\ \frac { 5 } { 36 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a positive constant.

1. Show that \(c = \frac { 1 } { 12 }\)
2. 1. Sketch the probability density function.
   2. Explain why the mode of \(X = 0\)
3. Find the cumulative distribution function of \(X\), for all values of \(x\)
4. Find, to 3 significant figures, the value of \(d\) such that \(\mathrm { P } ( X > d \mid X > 0 ) = \frac { 2 } { 5 }\)

   Leave blank	Q7

1. A random variable \(X\) has probability density function

$$f ( x ) = \begin{cases} \frac { 2 x } { 15 } & 0 \leqslant x \leqslant k \\ \frac { 1 } { 5 } ( 5 - x ) & k < x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Showing your working clearly, find the value of \(k\).
2. Write down the mode of \(X\).
3. Find \(\mathrm { P } \left( \left. X \leqslant \frac { k } { 2 } \right\rvert \, X \leqslant k \right)\)

5. A call centre records the length of time, \(T\) minutes, its customers wait before being connected to an agent. The random variable \(T\) has a cumulative distribution function given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 0 \\ 0.3 t - 0.004 t ^ { 3 } & 0 \leqslant t \leqslant 5 \\ 1 & t > 5 \end{array} \right.$$

1. Find the proportion of customers waiting more than 4 minutes to be connected to an agent.
2. Given that a customer waits more than 2 minutes to be connected to an agent, find the probability that the customer waits more than 4 minutes.
3. Show that the upper quartile lies between 2.7 and 2.8 minutes.
4. Find the mean length of time a customer waits to be connected to an agent.

1. The lifetime of a particular battery, \(T\) hours, is modelled using the cumulative distribution function

$$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 8 \\ \frac { 1 } { 96 } \left( 74 t - \frac { 5 } { 2 } t ^ { 2 } + k \right) & 8 \leqslant t \leqslant 12 \\ 1 & t > 12 \end{array} \right.$$

1. Show that \(k = - 432\)
2. Find the probability density function of \(T\), for all values of \(t\).
3. Write down the mode of \(T\).
4. Find the median of \(T\).
5. Find the probability that a randomly selected battery has a lifetime of less than 9 hours. A battery is selected at random. Given that its lifetime is at least 9 hours,
6. find the probability that its lifetime is no more than 11 hours.
   

1. At a cafe, customers ordering hot drinks order either tea or coffee.

Of all customers ordering hot drinks, \(80 \%\) order tea and \(20 \%\) order coffee. Of those who order tea, \(35 \%\) take sugar and of those who order coffee \(60 \%\) take sugar.

1. A random sample of 12 customers ordering hot drinks is selected. Find the probability that fewer than 3 of these customers order coffee.
2. 1. A randomly selected customer who orders a hot drink is chosen. Show that the probability that the customer takes sugar is 0.4
   2. Write down the distribution for the number of customers who take sugar from a random sample of \(n\) customers ordering hot drinks.
3. A random sample of 10 customers ordering hot drinks is selected.
   1. Find the probability that exactly 4 of these 10 customers take sugar.
   2. Given that at least 3 of these 10 customers take sugar, find the probability that no more than 6 of these 10 customers take sugar.
4. In a random sample of 150 customers ordering hot drinks, find, using a suitable approximation, the probability that at least half of them take sugar.

1. The time in minutes that Elaine takes to checkout at her local supermarket follows a continuous uniform distribution defined over the interval [3,9].

Find

1. Elaine's expected checkout time,
2. the variance of the time taken to checkout at the supermarket,
3. the probability that Elaine will take more than 7 minutes to checkout. Given that Elaine has already spent 4 minutes at the checkout,
4. find the probability that she will take a total of less than 6 minutes to checkout.
   

5. The continuous random variable \(T\) is used to model the number of days, \(t\), a mosquito survives after hatching. The probability that the mosquito survives for more than \(t\) days is $$\frac { 225 } { ( t + 15 ) ^ { 2 } } , \quad t \geqslant 0$$

1. Show that the cumulative distribution function of \(T\) is given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \frac { 225 } { ( t + 15 ) ^ { 2 } } & t \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the probability that a randomly selected mosquito will die within 3 days of hatching.
3. Given that a mosquito survives for 3 days, find the probability that it will survive for at least 5 more days. A large number of mosquitoes hatch on the same day.
4. Find the number of days after which only \(10 \%\) of these mosquitoes are expected to survive.