Decision Making

Key Terms

Stem: This is the information in which you are given. You may have to, infer information, draw conclusions, evaluate arguments, or analyse statistics in the stem.

Statement/Question: This is a statement or question which links to the stem and you must determine the answer of the question or validity of the statement.

In the decision making section, you have 29 questions to answer in 31 minutes, giving you an average of 64 seconds per question. However, as there are six question types, you will come to find some questions will take you shorter or longer than this. For most candidates, the timing for this section is more manageable than the verbal reasoning section. But remember: on tricker questions, make an educated guess, flag and skip. You can always come back at the end if you have time. Here at The Top Medic, we would complete around 24-25 questions on the first time round, and come back to the remaining questions if we had time. Almost all questions are worth one mark (the exceptions are explained below), so don’t let any question gobble up your valuable time!

There are 6 question types:

• Syllogisms: You are given a set of premises (statements) and must determine whether or not each conclusion (for a set of five conclusions) follows from the premises.

• Logical Puzzles: You are given a chunk of information and must determine which one of the four answer options is correct.

• Recognising Assumptions/Evaluating Arguments: You have to determine the strongest argument to support or contradict a solution to a problem.

• Interpreting Information: You are given a chunk of information in various formats (passage, graph, chart, table) and must determine whether or not each conclusion (for a set of 5 conclusions) follows from the information.

• Venn Diagrams: These questions can take various forms, including selecting the correct statement by interpreting a Venn diagram or selecting the correct Venn diagram based on a set of statements.

• Probabilistic and Statistical Reasoning: You have to calculate the probability of one or more outcomes by interpreting information from a passage. You must then use these calculations to select the best conclusion.

Syllogisms

As mentioned. You are given a set of premises (statements/facts) and must determine whether or not each conclusion (for a set of five conclusions) follows from the premises. The premises are always true and you must select:

• “Yes” if the conclusion follows from the premises or can be inferred.

• “No” if the conclusion doesn’t follow from the premises or cannot be inferred.

To answer these questions, read the premises first, focus on the relationships between subjects in the premise (If A = B, then A wears D, if A = C, then A wears E). It may help you to begin by phrasing these relationships in terms of letters. Then, read and scrutinise each conclusion. Each of these questions will give you 2 marks if you get all of the answers correct, and 1 mark if you get it partially correct.

Top Tips:

• Ensure the wording is the same as the premise: Often the conclusions are more specific or sweeping than the premise and so do not follow - e.g using all men must wear green or red, when the premises only focus on men in HiTown, or mentioning women when the premises only mentions men.

• Look out for qualifiers: This links to the tip above, but if the premise mentions “most”, and the conclusion states “few’, then it does not follow. In general: All = 100%, Most = 50% - 100%, Some = 0 - 100%, Minority = 0 - 50%, And a “few” is less than “some”.

• Be wary of “Yes”. They are keen to give you conclusions which seem like they may follow but do not. Be very cautious of the wording, and if the conclusion cannot be directly inferred, then it does not follow: “No”. Just because all of my pens are green, doesn’t mean all of my green things are pens (If X=Y, Y doesn’t necessarily always equal X).

Example: In HiTown, all men wear dresses. All the dresses are green, unless the man has brown eyes, and then he must wear a red dress.

1. No men in HiTown wear a yellow dresses

2. No women in HiTown wears a dress.

3. All men must wear a green or red dress.

A. Yes

The answers for the example:

1. A, Yes. The premises state “all men wear dresses.” and the dresses are “green” or “red”. Therefore, no man in HiTown wears a yellow dress.

2. B. No. This conclusion does not follow from the premises as they do not mention women. We cannot infer what the women wear from the premises so the conclusion doesn’t not follow.

3. B. No. This is a common trick the examiners like to use. This conclusion focuses on all men, rather than just men in HiTown (as stated in the premises). Therefore, we cannot make a sweeping conclusion about all men based on this subgroup. The conclusion does not follow.

Logical Puzzles

Here, you are given a chunk of information and must determine which one of the four conclusions is correct. These questions rely on logical reasoning and relationships between subjects. Ensure you read all the information given, as it is often all important. To answer these questions, we recommend:

• Use your whiteboard: These questions include many subjects making it very difficult to solve in your head. By using your whiteboard, it offloads information from your head and ensures you use all the information given.

• Use abbreviations for names or objects: Dave -> D, Eve -> E, Bullseye -> ✓

• Draw a diagram or table: As shown with the example, for almost all of these questions, a diagram (if the question is related to the position of objects e.g who sits next to who) or a table (for relationships between subjects) can help you solve them.

• You often don’t need to complete the puzzle: As there is only one question per logical puzzle, only partially solve the puzzle until you have the required answer, then move on! Completing the puzzle wastes time and often you are not provided with enough information to do so. In the Exam Style Question II below, it is impossible to completely solve the puzzle - so be careful!

Example: Dave, Eve, Fred and Grace are playing archery. One of them scores a bullseye. When asked who:

Dave says “it was Eve”

Eve says “It was Fred”

Fred says “It wasn’t Dave”

Grace says “It wasn’t Fred”

One of them is lying, who is it?

A. Dave

For this example, we would draw a table like the one shown. The header is the first letter of the name of each person, and the letters below are who said what. A tick indicates they scored the bullseye and a cross indicates they didn’t. Immediately, this table shows you that either Eve or Grace must be lying as there is a contradiction. If we refer back to the information, we will see only one bullseye was scored; therefore, Eve must have scored the bullseye (as Dave can’t be lying as there is only one liar). We know Fred did not score the bullseye and so Eve is lying. Option B. Eve is correct.

Recognising Assumptions/Evaluating Arguments:

In this question type, you are given a problem and a solution. You will then be given 4 arguments which either support or contradict that solution to the problem. You must select the argument which is the strongest. The strongest argument will:

• Have no unsupported assumptions - e.g the elderly prefer to drive slow - we don’t know this.

• Have no contradictions.

• Will mention the solution directly and how it relates to the problem directly (it will not just focus on a smaller subgroup e.g focusing on young people when the stem mentions people).

• May add supporting evidence.

• Be an argument and not just a statement.

When answering this question type, identify the problem and the solution in the stem. Then read each answer, noting the letters of the answers which mention both the problem and the solution. Once you have identified these, choose the strongest argument using the criteria listed above, particularly the argument which adds evidence and avoids assumptions.

Example: Should we have a minimum speed limit in order to improve road safety? Select the strongest argument from the statements below.

A. Yes, A minimum speed limit will ensure there are no slow drivers, reducing rush hour traffic.

B. Yes, 40% of crashes are due to slow drivers. By legalising a minimum speed limit, we can prevent these crashes.

C. No, some drivers ignore the maximum speed limit, hence, many drivers will do the same with a minimum speed limit.

D. No, a minimum speed limit will encourage people to drive faster than they are comfortable with. This will deter people from driving, gatekeeping it from the elderly.

For the example, you will have identified “improving road safety” as the problem and “minimum speed limit” as the solution. Option B. is the only option which mentions both the problem and the solution and, therefore, is the strongest argument. In the exam, you would select B. and move on, but we will discuss why the other options are not as suitable:

A. This talks about reducing rush hour traffic, which is not the same problem as that mentioned in the stem (road safety). In order to support the stem, we must assume that rush hour traffic results in more crashes, and as this is an unsupported assumption, it is a weak argument.

B. This is the strongest argument. This is because it mentions both the problem and the solution as well as providing evidence to support the claim (40% of crashes are due to slow drivers).

C. This argument highlights limitations of a minimum speed limit. However, it only talks about “some” drivers and, thus, doesn’t argue why it still wouldn’t be beneficial to have a minimum speed limit (rather it discusses why it would be less than optimal). Notice this option focuses on a smaller subgroup: “some drivers”, than the stem “all drivers”. Moreover, it doesn’t mention road safety. It is a weaker argument.

D. This argument does not mention road safety, and instead relies on two assumptions: 1. Elderly people drive slower 2. Faster cars result in more crashes. Even if you believe these assumptions to be correct, they are unsupported and so this option is a weak argument.

Interpreting Information:

You are given a chunk of information in various formats (passage, graph, chart, table) and must determine whether or not each conclusion (for a set of five conclusions) follows from the information. Remember from the syllogisms section:

• “Yes” if the conclusion follows from the premises or can be inferred.

• “No” if the conclusion doesn’t follow from the premises or cannot be inferred.

The majority of mistakes are made by not reading column headings or graph axis correctly. Common tricks the examiners use include:

• Units: each unit on a graph may represent £1000; read the axis (£1000s). Alternatively the graph may be recorded in kilometers but the conclusion is in metres.

• Time: the graph may be per day but the conclusion asks per year.

• Sales ≠ Profit, Sales ≠ Cost, and Items sold ≠ Profit.

• Look out for negatives and qualifiers in the conclusions.

Furthermore, always read the stem before and after the data, these may provide crucial bits of information. As with the syllogisms section, all five answers correct will gain 2 marks, and a partially correct answer will gain you 1 mark.

Example: Below, the graph shows the changes to the cost of living in the UK from 1987 -2022.

Select “Yes” if the conclusion does follow. Select “No” if the conclusion does not follow.

1. The cost of living has increased year on year from 1987 to 2000.

A. Yes

B. No

2. From 1987 to 2022, the cost of living has increased by roughly 135%.

3. The decrease in the cost of living in 2011 was due to the Scottish National Party winning a majority.

4. In 2000, the cost of living annually for a family of 4 would be roughly £48000.

In the example:

1. B. No. This is false as we can see the cost of living decreased in 1994.

2. A. Yes. This is true. In 1987, the cost of living was roughly £900. In 2022, the cost of living is roughly £2100. Therefore, the percentage increase is given by 100 x (2100-900)/900 = 133%. This is roughly 135% and so the conclusion does follow.

3. B No. Although you can see there was a decrease in the cost of living at 2011, the information gives us no indication of the cause. The conclusion cannot be inferred from the information.

4. A Yes. In 2000, the cost of living per person per month was roughly £1,000. To calculate the cost of living per person annually: 12 x £1,000 = £12,000. The cost of living of a family of 4 will be 4 x £12,000 = £48000. The conclusion does follow from the information.

Venn Diagrams:

These questions can take three main forms. Interpreting a Venn diagram, understanding information in order to produce a Venn diagram, or using a Venn diagram to help you solve a problem. These questions often take a shorter time then the others and are located near the end of this section. So provided you guessed, flag, and skipped appropriately, you will always have time to complete these questions.

For interpreting Venn diagram questions read the text first, then the question, and finally the diagram. Focus only on the relevant shapes.

• Be careful of overlaps, double check to ensure the value you have chosen is between (or outside) the overlaps of the required shapes.

• You may have to add or subtract numbers or probabilities. Ensure you take time to read the question.

• You may have to calculate percentages. For this you need to know the total number of people/subjects and the number of people/subjects doing the event of interest.

• Be careful of “amongst…” or “given that…” questions. For these, the total number of people/subjects is no longer everyone, but rather everyone doing the event specified (see the example for more details).

For producing Venn Diagram questions:

• The sum of the numbers inside the Venn diagram must be the total of everything.

• Look at the diagrams to see the position you should place numbers when the statement includes “only”.

• Use logic to deduce the other numbers. The sum of all the numbers in set A should equal the total number of events in set A. The same is true for all the other sets.

Finally, for the hidden Venn diagram questions: These will neither have the question or answer in the form of a Venn diagram, but rather, drawing a Venn diagram on your whiteboard will help you answer the question. The stem for these questions will usually include statements such as “only A”, “only B”, “A and B” and “Neither A nor B”, where A and B can be any subject. These questions become easier to identify with practice. See Exam Style Question V for more details.

This is the main equation you will use for Venn Diagram questions. To convert from probability to a percentage, multiply by 100. To explain this equation, let’s say we are interested in the probability that a student plays only football:

Frequency: “The number of times an event has occurred.” The frequency of people who play only football - top of the triangle - is 21. We are not interested in the other numbers in the triangle, as these have multiple overlaps and, therefore, play sports other than football.

Event: This is the outcome/set of outcomes we are interested in. For this example it is playing football only. Note this can be more complex e.g. playing hockey and tennis but not football. It is crucial you check the overlaps.

Set: This is the group in which we are interested in. The set is either everyone (like in the example above where we want to see the probability that a student plays football out of the 114 students - the sum of the frequencies of all the events in the Venn diagram) or it may focus on a specific group, these will be expressed as “amongst…” or “given that…” questions (see below).

Consequently, using the equation, the probability that someone plays only football is 21/114 = 0.184 = 18.4%

However, if the question was “given that the student plays football, what is the probability that they play tennis as their only other sport.” Here, the set is the people who play football (as we know that the students we are selecting from all play football) and so we must total the frequencies of all the students who play football - the sum of the frequencies in the triangle: 21 + 12 + 17 + 1 + 3 + 2 + 11 + 9 + 19 + 4 = 99.

The frequency of the event occurring is the frequency of people who play football and rugby only (triangle and tear only) which is 2. Using the equation, the probability that they play only rugby given that they play football is therefore 2/99= 0.02. This is equal to 2%

Example: A school conducted a survey on 114 students about which school sports clubs they were involved in.

Amongst the students who played badminton, what percentage of these students also played rugby and football but not hockey?

A. 2.7%

For the example question, “Amongst the students who played badminton, what percentage of these students also played rugby and football but not hockey?” You will have noticed this is an amongst question; the set is “people who play badminton”. The total number of people who play badminton (bottom number of the equation) is the sum of all the numbers in the heart: 9 + 1 + 3 + 11 = 24.

The frequency of the event (play badminton, rugby and football but not hockey) is where the heart, tear and triangle overlap, but it is outside the square. This is 3. Therefore, the probability “amongst the students who played badminton, students also played rugby and football but not hockey” is 3/24 = 0.125. This is equal to 12.5%. The correct answer is C.

Probabilistic and Statistical Reasoning:

As previously mentioned, this section involves calculating the probability of one or more outcomes by interpreting information from a passage. You must then use this/these calculation(s) to select the best conclusion.

The probability calculations in this section are GCSE level probability and are explained above in the Venn diagram section. The equation is the same as before. The only additional fact you need to know is:

Let the probability of A be 0.2 and B be 0.3

• And = multiply probabilities. The probability of A and B = 0.2 x 0.3 = 0.06. There is a 0.06 probability that A and B occurs.

• Or = add probabilities. The probability of A or B = 0.2 + 0.3 = 0.5. There is a 0.5 probability that A or B occurs.

You can divide probabilistic and statistical reasoning questions into two steps, the maths step and the logical reasoning step. For the maths step:

• If a question talks about “A and B,” and “neither A nor B” often a Venn diagram can help you out.

• If a questions requires the probability of a combination, this is often a rather short combination. In this case, write out the possible options (e.g if a lock can be up to three letters long, non-order dependant, of A and B: A, B, AA, BB, AB, AAA, AAB, ABB, BBB).

• If the question involves conditional probability, draw a tree diagram .

Conditional probability is when the probability of an event changes depending on the events before it, e.g. in the example, each go you remove the marble. Therefore, if you remove a green marble on the first go, the probability of pulling out a green on the second go falls from 5/12 to 4/11 (as there is only 4 greens left, but still 7 blues remaining).

For the logical reasoning step:

• This is often made clear by the maths. When you have calculated the probability of the event, choose the argument which doesn’t contradict itself, mentions the probability you have calculated, and doesn’t make any foolish assumptions (e.g. if the probability is 0.9, this is still not “always”).

Example: A game costs £1 a go. In a bag of twelve marbles, five are green and seven are blue. If you pull out two green marbles in a row (removing them from the bag each time) you win £10. Should Susan play the game?

A. Yes, The probability of winning is 0.15, Therefore in 10 attempts, its likely that Susan will win, and thus make a profit.

B. Yes, the probability of winning is above 0.10. Therefore after 10 goes, Susan will certainly have won, making a profit

C. No, the probability of pulling out two green marbles sequentially is less than 0.10. It is unlikely Susan will win within 10 goes, and thus she will likely make a loss.

D. No, There are more blue marbles than green marbles. It is very unlikely Susan will ever win.

For the example, the correct answer is A.

The first section involves a conditional probability problem, and thus you may have drawn a tree diagram like the one we drew. Note, we have not completed it, as the problem is only interested in drawing two green marbles, so to calculate the probability of the other branches is a waste of time. As you have to draw one green marble and another green marble in order to win, we need to multiply the probabilities. Therefore, drawing two green marbles = 5/12 x 4/11 = 0.15. Instantly, this tells us conclusion C is wrong.

If you struggle with calculating probabilities, on the tree diagram we have also visualised the marbles in the bag. As you can see, on the first draw there are 5 green and 7 blue, so the probability of drawing a green is 5 out of (5 + 7 = ) 12 marbles: 5/12. For the second draw, we have lost a green marble, and hence there are now only 11 marbles in total (4 + 7 = 11). As we have removed one green marble, there are only 4 green marbles left in the bag, so for the second draw, the probability of pulling a green is 4/11. Note, the method of calculating probability is the same as that with Venn diagrams: in this case the “event” = pulling a green, and the “frequency of all events in a set” = the number of marbles.

Answer B is also wrong, as it states after 10 goes it is certain Susan will win. This is not the case as you simply cannot add 0.15 ten times. A probability of more than 1.0 is impossible (For the A-level mathematicians: this is because to calculate the probability of Susan winning on her second attempt, we must include the probability that she failed the first time: 0.85 x 0.15, therefore, this can be modelled as the binomial distribution, where it tends to, but never reaches 1.0). Therefore, repeated probabilities are actually more complicated than simply adding it all up. In an exam we obviously wouldn’t calculate this, however, always be cautious of “certain”; unless the probability is 1.0, the outcome is never certain, regardless of how many times you repeat it. Option D. is not correct because it provides no mathematical reasoning, and it makes the incorrect assumption “its unlikely Susan will ever win”. This is because after enough goes, the probability that Susan will win approaches 1.0 and in fact it will become very likely!