Conditional Probability

167 questions · 29 question types identified

Basic two-way table probability

Questions that present a complete two-way table with all values given and ask for simple probabilities or conditional probabilities directly from the table using P(A∩B)/P(B) or similar.

21
12.6% of questions
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3 The table shows the colour of hair and the colour of eyes of a sample of 750 people from a particular population.
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Given conditional, find joint or marginal

A question is this type if and only if it provides conditional probabilities P(A|B) and asks to find joint probability P(A∩B) or marginal probabilities using the multiplication rule.

14
8.4% of questions
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2. Events \(A\) and \(B\) are such that \(P ( A \cup B ) = 0.95 , P ( A \cap B ) = 0.6\) and \(P ( A \mid B ) = 0.75\).
i. Find \(P ( B )\).
ii. Find \(P ( A )\).
iii. Show that the events \(A ^ { \prime }\) and \(B\) are independent.
[0pt] [BLANK PAGE]
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Standard Bayes with discrete events

Questions where P(A|B) is found using Bayes' theorem with discrete events and a simple two-stage tree diagram (one initial choice, one outcome), typically involving 2-4 initial categories.

14
8.4% of questions
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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.
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Independence test requiring preliminary calculations

Questions where P(A), P(B), or P(A∩B) must first be calculated from given information (such as P(A∪B), Venn diagram data, or other relationships) before testing independence.

11
6.6% of questions
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2. Given that \(\mathrm { P } ( A ) = \frac { 3 } { 5 } , \mathrm { P } ( B ) = \frac { 5 } { 8 } , \mathrm { P } ( A \cap B ) = \frac { 7 } { 20 } , \mathrm { P } ( A \cup C ) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid A ) = \frac { 1 } { 3 }\),
  1. determine whether \(A\) and \(B\) are independent events.
  2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
  3. Find \(\mathrm { P } \left( ( A \cap C ) ^ { \prime } \right)\).
  4. Find \(\mathrm { P } ( A \mid C )\).
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Dice/random device selects population

A random device (dice, spinner, etc.) is used to determine which population or group to sample from, requiring probability tree with initial branching based on device outcome.

10
6.0% of questions
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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.
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Tree diagram with two-stage events

A question is this type if and only if it involves drawing or using a tree diagram to represent sequential or two-stage events (e.g., mode of transport then arrival time, coin toss then marble selection).

9
5.4% of questions
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  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 \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\).
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Venn diagram with two events

Questions involving exactly two events (typically labeled with two letters like A and B, or W and L, or G and R) where students must find probabilities of unions, intersections, complements, or conditional probabilities using a two-circle Venn diagram.

9
5.4% of questions
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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.
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Standard two-outcome diagnostic test

Questions involving a diagnostic test with exactly two outcomes (positive/negative) and requiring calculation of probability of disease given test result using Bayes' theorem.

9
5.4% of questions
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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 ?
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Conditional probability with algebraic expressions

A question is this type if and only if it requires manipulating algebraic expressions involving probabilities (often with variables like x, p, q) to find relationships or specific values.

8
4.8% of questions
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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 \(x\).
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Finding unknown probability from total probability

A question is this type if and only if it requires finding an unknown probability value (often denoted x, p, or m) by using the law of total probability or complement rule with given conditional probabilities.

7
4.2% of questions
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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 \(x\).
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Bayes with complementary outcome

Questions specifically asking for P(A|B') where B' is the complement of B (e.g., given NOT late, given does NOT have coffee, given does NOT have jam), requiring calculation of P(B') first.

7
4.2% of questions
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(ii) Given that Nikita's mother does not like her present, find the probability that the present is a scarf.
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Venn diagram with three events

Questions involving exactly three events (typically three sports, three methods, or three categories) where students must interpret or complete a three-circle Venn diagram to find probabilities or counts in various regions.

6
3.6% of questions
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  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. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\)
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Conditional probability with given score/outcome

A question is this type if and only if it involves finding the probability of a prior event given a particular observed score or outcome (e.g., given score is 12, find probability spinner landed on even).

5
3.0% of questions
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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.
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Multi-stage game or match outcomes

A question is this type if and only if it involves a game or match with multiple rounds where the probability of winning each round depends on previous outcomes.

5
3.0% of questions
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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.
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Incomplete two-way table completion

Questions that provide partial information about a two-way table (some cells missing, or probabilities given instead of frequencies) and require completing the table before calculating probabilities.

5
3.0% of questions
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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.
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Sampling without replacement from bags/boxes

A question is this type if and only if it involves selecting multiple items sequentially without replacement from a container with different colored or numbered objects.

4
2.4% of questions
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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.
(a)Find the probability that the green bead is one of the beads selected. The probability that both of the beads are red is \(\frac { 5 } { 84 }\)
(b)Show that \(r\) satisfies the equation \(r ^ { 2 } - r - 240 = 0\)
(c)Hence show that the only possible value of \(r\) is 16
(d)Given that at least one of the beads is red,find the probability that they are both red.
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Probability distribution from conditional setup

A question is this type if and only if it requires constructing a complete probability distribution table for a random variable defined in a conditional probability context.

4
2.4% of questions
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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 \(X\).
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Probability with replacement vs without replacement

A question is this type if and only if it explicitly contrasts or involves scenarios where items are either replaced or not replaced between selections.

4
2.4% of questions
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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.
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Complement and union/intersection laws

A question is this type if and only if it requires using complement rules, De Morgan's laws, or addition rules to find probabilities of unions, intersections, or complements.

2
1.2% of questions
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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.
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Conditional with three or more stages

A question is this type if and only if it involves three or more sequential stages or decisions where probabilities at each stage depend on previous outcomes.

2
1.2% of questions
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6 Deeti has 3 red pens and 1 blue pen in her left pocket and 3 red pens and 1 blue pen in her right pocket. 'Operation \(T\) ' consists of Deeti taking one pen at random from her left pocket and placing it in her right pocket, then taking one pen at random from her right pocket and placing it in her left pocket.
  1. Find the probability that, when Deeti carries out operation \(T\), she takes a blue pen from her left pocket and then a blue pen from her right pocket. The random variable \(X\) is the number of blue pens in Deeti's left pocket after carrying out operation \(T\).
  2. Find \(\mathrm { P } ( X = 1 )\).
  3. Given that the pen taken from Deeti's right pocket is blue, find the probability that the pen taken from Deeti's left pocket is blue.
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Independence test with P(A∩B) = P(A)×P(B)

Questions that provide P(A), P(B), and P(A∩B) directly and ask to test independence using the product rule P(A∩B) = P(A)×P(B).

2
1.2% of questions
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14
  1. \(\quad\) Find \(\mathrm { P } ( A )\)
    14
  2. \(\quad\) Find \(\operatorname { P } ( B \mid A )\)
    14
  3. Determine if \(A\) and \(B\) are independent events.
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Independence test with conditional probability

Questions that provide P(A|B) or require calculating it, and test independence by checking if P(A|B) = P(A) or equivalent conditional probability relationships.

2
1.2% of questions
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4 Each weekday, Marta travels to school by bus. Sometimes she arrives late.
  • \(L\) is the event that Marta arrives late.
  • \(R\) is the event that it is raining.
You are given that \(\mathrm { P } ( L ) = 0.15 , \mathrm { P } ( R ) = 0.22\) and \(\mathrm { P } ( L \mid R ) = 0.45\).
  1. Use this information to show that the events \(L\) and \(R\) are not independent.
  2. Find \(\mathrm { P } ( L \cap R )\).
  3. Draw a Venn diagram showing the events \(L\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
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Person selected from combined populations

A single person or item is selected at random from the combined total of all populations, requiring weighted probabilities based on population sizes or proportions.

2
1.2% of questions
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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.
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Testing mutual exclusivity

A question is this type if and only if it asks to determine whether events are mutually exclusive by checking if P(A∩B) = 0.

1
0.6% of questions
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  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 \(s\)
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Exactly k successes in n trials

A question is this type if and only if it asks for the probability of exactly k occurrences of an event in n trials, requiring enumeration of specific sequences.

1
0.6% of questions
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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.
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Bayes with sampling without replacement

Questions involving conditional probability after selecting multiple items without replacement from containers (boxes, bags), where probabilities change after first selection.

1
0.6% of questions
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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 \(A\).
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Three-outcome diagnostic test

Questions involving a diagnostic test with three outcomes (positive/doubtful/negative) requiring extended conditional probability calculations.

1
0.6% of questions
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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.
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Multi-type disease diagnostic

Questions involving a disease with multiple mutually exclusive types (e.g., type A and type B) where diagnostic characteristics differ by type.

1
0.6% of questions
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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.
    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.
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At least one success in repeated trials

A question is this type if and only if it asks for the probability of at least one occurrence of an event across multiple independent or dependent trials.

0
0.0% of questions