Conditional Probability

3 The table shows the colour of hair and the colour of eyes of a sample of 750 people from a particular population.

3 Ronnie obtained data about the gross domestic product (GDP) and the birth rate for 170 countries. He classified each GDP and each birth rate as either 'low', 'medium' or 'high'. The table shows the number of countries in each category. One of these countries is chosen at random.

1. Find the probability that the country chosen has a medium GDP.
2. Find the probability that the country chosen has a low birth rate, given that it does not have a medium GDP.
3. State with a reason whether or not the events 'the country chosen has a high GDP' and 'the country chosen has a high birth rate' are exclusive. One country is chosen at random from those countries which have a medium GDP and then a different country is chosen at random from those which have a medium birth rate.
4. Find the probability that both countries chosen have a medium GDP and a medium birth rate.

1. A company has 1825 employees.

The employees are classified as professional, skilled or elementary.
The following table shows

• the number of employees in each classification
• the two areas, \(A\) or \(B\), where the employees live

An employee is chosen at random.
Find the probability that this employee

1. is skilled,
2. lives in area \(B\) and is not a professional. Some classifications of employees are more likely to work from home.
   • \(65 \%\) of professional employees in both area \(A\) and area \(B\) work from home
   • \(40 \%\) of skilled employees in both area \(A\) and area \(B\) work from home
   • \(5 \%\) of elementary employees in both area \(A\) and area \(B\) work from home
   • Event \(F\) is that the employee is a professional
   • Event \(H\) is that the employee works from home
   • Event \(R\) is that the employee is from area \(A\)
   • Using this information, complete the Venn diagram on the opposite page.
   • Find \(\mathrm { P } \left( R ^ { \prime } \cap F \right)\)
   • Find \(\mathrm { P } \left( [ H \cup R ] ^ { \prime } \right)\)
   • Find \(\mathrm { P } ( F \mid H )\)
   \includegraphics[max width=\textwidth, alt={}]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-13_872_1020_294_525}
   Turn over for a spare diagram if you need to redraw your Venn diagram. Only use this diagram if you need to redraw your Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-15_872_1017_392_525}

5 In a certain country \(54 \%\) of the population is male. It is known that \(5 \%\) of the males are colour-blind and \(2 \%\) of the females are colour-blind. A person is chosen at random and found to be colour-blind. By drawing a tree diagram, or otherwise, find the probability that this person is male.

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.

7 Each question on a multiple-choice examination paper has \(n\) possible responses, only one of which is correct. Joni takes the paper and has probability \(p\), where \(0 < p < 1\), of knowing the correct response to any question, independently of any other. If she knows the correct response she will choose it, otherwise she will choose randomly from the \(n\) possibilities. The events \(K\) and \(A\) are 'Joni knows the correct response' and 'Joni answers correctly' respectively.

1. Show that \(\mathrm { P } ( A ) = \frac { q + n p } { n }\), where \(q = 1 - p\).
2. Find \(P ( K \mid A )\). A paper with 100 questions has \(n = 4\) and \(p = 0.5\). Each correct response scores 1 and each incorrect response scores - 1 .
3. (a) Joni answers all the questions on the paper and scores 40 . How many questions did she answer correctly?
   (b) By finding the distribution of the number of correct answers, or otherwise, find the probability that Joni scores at least 40 on the paper using her strategy.

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. \(\mathrm { P } ( B )\).

4 A supermarket has a large stock of eggs. 40\% of the stock are from a firm called Eggzact. 12\% of the stock are brown eggs from Eggzact. An egg is chosen at random from the stock. Calculate the probability that

1. this egg is brown, given that it is from Eggzact,
2. this egg is from Eggzact and is not brown.

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 \(X\).

1 When Shona goes to college she either catches the bus with probability 0.8 or she cycles with probability 0.2 . If she catches the bus, the probability that she is late is 0.4 . If she cycles, the probability that she is late is \(x\). The probability that Shona is not late for college on a randomly chosen day is 0.63 . Find the value of \(x\).

2 Maria has 3 pre-set stations on her radio. When she switches her radio on, there is a probability of 0.3 that it will be set to station 1, a probability of 0.45 that it will be set to station 2 and a probability of 0.25 that it will be set to station 3 . On station 1 the probability that the presenter is male is 0.1 , on station 2 the probability that the presenter is male is 0.85 and on station 3 the probability that the presenter is male is \(p\). When Maria switches on the radio, the probability that it is set to station 3 and the presenter is male is 0.075 .

1. Show that the value of \(p\) is 0.3 .
2. Given that Maria switches on and hears a male presenter, find the probability that the radio was set to station 2.

A sample of 200 households was obtained from a small town. Each household was asked to complete a questionnaire about their purchases of takeaway food. \(A\) is the event that a household regularly purchases Indian takeaway food. \(B\) is the event that a household regularly purchases Chinese takeaway food. It was observed that \(P(B|A) = 0.25\) and \(P(A|B) = 0.1\) Of these households, 122 indicated that they did not regularly purchase Indian or Chinese takeaway food. A household is selected at random from those in the sample. Find the probability that the household regularly purchases both Indian and Chinese takeaway food. [6 marks]

2 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.

• \(W\) is the event that this person speaks Welsh.
• \(C\) is the event that this person is a child.

You are given that \(\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17\) and \(\mathrm { P } ( W \cap C ) = 0.06\).

1. Determine whether the events \(W\) and \(C\) are independent.
2. Draw a Venn diagram, showing the events \(W\) and \(C\), and fill in the probability corresponding to each region of your diagram.
3. Find \(\mathrm { P } ( W \mid C )\).
4. Given that \(\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169\), use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults.

6. For the events \(A\) and \(B\), $$\mathrm { P } \left( A \cap B ^ { \prime } \right) = 0.32 , \mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.11 \text { and } \mathrm { P } ( A \cup B ) = 0.65$$

1. Draw a Venn diagram to illustrate the complete sample space for the events \(A\) and \(B\).
2. Write down the value of \(\mathrm { P } ( A )\) and the value of \(\mathrm { P } ( B )\).
3. Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
4. Determine whether or not \(A\) and \(B\) are independent.

3 A hotel has three types of room: double, twin and suite. The percentage of rooms in the hotel of each type is 40,45 and 15 respectively. Each room in the hotel may be occupied by \(0,1,2\), or 3 or more people. The proportional occupancy of each type of room is shown in the table.

2 A hill-top monument can be visited by one of three routes: road, funicular railway or cable car. The percentages of visitors using these routes are 25, 35 and 40 respectively. The age distribution, in percentages, of visitors using each route is shown in the table. For example, 15 per cent of visitors using the road were under 18 . Calculate the probability that a randomly selected visitor:

1. who used the road is aged 18 or over;
2. is aged between 18 and 64;
3. used the funicular railway and is aged over 64;
4. used the funicular railway, given that the visitor is aged over 64.

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]

2 During an outbreak of a disease, it is known that \(68 \%\) of people do not have the disease. Of people with the disease, \(96 \%\) react positively to a test for diagnosing it, as do \(m \%\) of people who do not have the disease.

1. In the case \(m = 8\), find the probability that a randomly chosen person has the disease, given that the person reacts positively to the test.
2. What value of \(m\) would be required for the answer to part (i) to be 0.95 ?

1. The following incomplete tree diagram shows the relationships between the event \(A\) and the event \(B\). \includegraphics[max width=\textwidth, alt={}, center]{77ae01cd-2b58-48ab-889f-272e27ecf99d-14_799_839_351_548}

Given that \(\mathrm { P } ( B ) = \frac { 9 } { 20 }\)

1. find \(\mathrm { P } ( A )\) and complete the tree diagram,
2. find \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\).

8 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3\) and \(\mathrm { P } ( A \mid B ) = 0.6\).

1. Show that \(\mathrm { P } ( B ) \leqslant 0.5\).
2. Given also that \(\mathrm { P } ( A \cup B ) = x\), find \(\mathrm { P } ( B )\) in terms of \(x\).

3. In a study of 120 pet-owners it was found that 57 owned at least one dog and of these 16 also owned at least one cat. There were 35 people in the group who didn't own any cats or dogs. As an incentive to take part in the study, one participant is chosen at random to win a year's free supply of pet food. Find the probability that the winner of this prize

1. owns a dog but does not own a cat,
2. owns a cat,
3. does not own a cat given that they do not own a dog.

4 At a sixth form college, the student council has 16 members made up as follows. There are 3 male and 3 female students from Year 12, and 6 male and 4 female students from Year 13. Two members of the council are chosen at random to represent the college at conference. Find the probability that the 2 members chosen are

1. the same sex,
2. the same sex and from the same year,
3. from the same year given that they are the same sex.

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.

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

3 A fair five-sided spinner has sides numbered 1,2,3,4,5. Raj spins the spinner and throws two fair dice. He calculates his score as follows.

• If the spinner lands on an even-numbered side, Raj multiplies the two numbers showing on the dice to get his score.
• If the spinner lands on an odd-numbered side, Raj adds the numbers showing on the dice to get his score.

Given that Raj's score is 12, find the probability that the spinner landed on an even-numbered side.

3 Roger and Andy play a tennis match in which the first person to win two sets wins the match. The probability that Roger wins the first set is 0.6 . For sets after the first, the probability that Roger wins the set is 0.7 if he won the previous set, and is 0.25 if he lost the previous set. No set is drawn.

1. Find the probability that there is a winner of the match after exactly two sets.
2. Find the probability that Andy wins the match given that there is a winner of the match after exactly two sets.

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]

7 Dayo chooses two digits at random, without replacement, from the 9-digit number 113333555.

1. Find the probability that the two digits chosen are equal.
2. Find the probability that one digit is a 5 and one digit is not a 5 .
3. Find the probability that the first digit Dayo chose was a 5, given that the second digit he chose is not a 5 .
4. The random variable \(X\) is the number of 5s that Dayo chooses. Draw up a table to show the probability distribution of \(X\).

7. In a large college, \(\frac { 3 } { 5 }\) of the students are male, \(\frac { 3 } { 10 }\) of the students are left handed and \(\frac { 1 } { 5 }\) of the male students are left handed. A student is chosen at random.

1. Given that the student is left handed, find the probability that the student is male.
2. Given that the student is female, find the probability that she is left handed.
3. Find the probability that the randomly chosen student is male and right handed. Two students are chosen at random.
4. Find the probability that one student is left handed and one is right handed.

Two events \(A\) and \(B\) are such that \(\text{P}(A) = 0.6\), \(\text{P}(B) = 0.5\) and \(\text{P}(A \cup B) = 0.85\). Find \(\text{P}(A | B)\). [4]

2 Boxes of sweets contain toffees and chocolates. Box \(A\) contains 6 toffees and 4 chocolates, box \(B\) contains 5 toffees and 3 chocolates, and box \(C\) contains 3 toffees and 7 chocolates. One of the boxes is chosen at random and two sweets are taken out, one after the other, and eaten.

1. Find the probability that they are both toffees.
2. Given that they are both toffees, find the probability that they both came from box \(A\).

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]

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.

3. The probability that Ajita gets up before 6.30 am in the morning is 0.7 The probability that she goes for a run in the morning is 0.35
The probability that Ajita gets up after 6.30 am and does not go for a run is 0.22
Let \(A\) represent the event that Ajita gets up before 6.30 am and \(B\) represent the event that she goes for a run in the morning. Find

1. \(\mathrm { P } ( A \cup B )\),
2. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
3. \(\mathrm { P } ( B \mid A )\).
4. State, with a reason, whether or not events \(A\) and \(B\) are independent.

In a large examination room each candidate has just one electronic calculator.

• \(G\) is the event that a randomly chosen candidate has a graphical calculator.
• \(T\) is the event that a randomly chosen candidate has a 'Texio' brand calculator.

You are given the following probabilities. $$\text{P}(G) = 0.65 \quad \text{P}(T) = 0.4 \quad \text{P}(G \cap T) = 0.25$$

1. Are the events \(G\) and \(T\) independent? Justify your answer with an appropriate calculation. [2]
2. Find P(\(T | G\)) and explain, in the context of this question, what this probability represents. [3]

2 In country \(X , 25 \%\) of people have fair hair. In country \(Y , 60 \%\) of people have fair hair. There are 20 million people in country \(X\) and 8 million people in country \(Y\). A person is chosen at random from these 28 million people.

1. Find the probability that the person chosen is from country \(X\).
2. Find the probability that the person chosen has fair hair.
3. Find the probability that the person chosen is from country \(X\), given that the person has fair hair.

1. The Venn diagram shows the probabilities associated with four events, \(A , B , C\) and \(D\) \includegraphics[max width=\textwidth, alt={}, center]{2b63aa7f-bc50-4422-8dc0-e661b521c221-02_505_861_296_602}
   1. Write down any pair of mutually exclusive events from \(A , B , C\) and \(D\)
   Given that \(\mathrm { P } ( B ) = 0.4\)
2. find the value of \(p\) Given also that \(A\) and \(B\) are independent
3. find the value of \(q\) Given further that \(\mathrm { P } \left( B ^ { \prime } \mid C \right) = 0.64\)
4. find
   1. the value of \(r\)
   2. the value of \(s\)

6 Vehicles approaching a certain road junction from town \(A\) can either turn left, turn right or go straight on. Over time it has been noted that of the vehicles approaching this particular junction from town \(A\), \(55 \%\) turn left, \(15 \%\) turn right and \(30 \%\) go straight on. The direction a vehicle takes at the junction is independent of the direction any other vehicle takes at the junction.

1. Find the probability that, of the next three vehicles approaching the junction from town \(A\), one goes straight on and the other two either both turn left or both turn right.
2. Three vehicles approach the junction from town \(A\). Given that all three drivers choose the same direction at the junction, find the probability that they all go straight on.

7 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, \(3 \%\) is classified as positive, \(6 \%\) as doubtful and the rest negative. In fact, of the people who test positive, only \(95 \%\) have the disease. Of the people who test doubtful, \(10 \%\) have the disease. Of the people who test negative, \(1 \%\) actually have the disease. People who do not have the disease are described as 'clear'.

1. Copy and complete the tree diagram to show this information. \includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-5_830_1157_845_536}
2. Find the probability that a randomly selected person tests negative and is clear.
3. Find the probability that a randomly selected person has the disease.
4. Find the probability that a randomly selected person tests negative given that the person has the disease.
5. Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in \(98 \%\) of cases. If a person is clear, the examination will always correctly identify this.
6. A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.

3. A certain disease occurs in a population in 2 mutually exclusive types. It is difficult to diagnose people with type \(A\) of the disease and there is an unknown proportion \(p\) of the population with type \(A\).
It is easier to diagnose people with type \(B\) of the disease and it is known that \(2 \%\) of the population have type \(B\). A test has been developed to help diagnose whether or not a person has the disease. The event \(T\) represents a positive result on the test. After a large-scale trial of the test, the following information was obtained. For a person with type \(B\) of the disease the probability of a positive test result is 0.96 For a person who does not have the disease the probability of a positive test result is 0.05 For a person with type \(A\) of the disease the probability of a positive test result is \(q\)

1. Complete the tree diagram. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-08_776_965_1050_484} The probability of a randomly selected person having a positive test result is 0.169 For a person with a positive test result, the probability that they do not have the disease is \(\frac { 41 } { 169 }\)
2. Find the value of \(p\) and the value of \(q\). A doctor is about to see a person who she knows does not have type \(B\) of the disease but does have a positive test result.
3. 1. Find the probability that this person has type \(A\) of the disease.
   2. State, giving a reason, whether or not the doctor will find the test useful.