Probabilistic Reasoning

These questions give you a scenario with numerical data - counters in a bag, success rates of machines, test results - and ask which option is most/least likely, or whether a probability claim is valid. They have 4 options and are worth 1 mark.

They make up ~4-5 questions (~11.1% of DM). Target time: 15-30 seconds for simple questions, 45-60 seconds for combined events.

Most PR questions ask you to compare options. The table makes that comparison mechanical.

Step 1: Identify the variables. What's being compared? What "considering only" factors does the question specify?

Step 2: Build a quick table. Options as rows, variables as columns. Fill in the numbers from the stimulus.

Step 3: Normalise the units. All values must be in the same format. Convert failure rates to success rates, fractions to percentages, or whatever gives you a like-for-like comparison.

Step 4: Find the winner. One option usually dominates. If two are close, calculate more precisely.

Step 5: Re-read the question. Did they ask for MOST likely or LEAST likely? Highest or lowest? This catches reversal errors - and they're more common than you'd think.

You don't need advanced probability. These five principles cover every PR question in the UCAT.

This is the single most common trick in PR. The question presents numbers in deliberately mismatched formats so a quick glance gives the wrong answer.

Robot A: 78% success rate
Robot B: 4% failure rate

Quick glance: "78 > 4, so A is better." - wrong.
Convert to the same unit: Robot B's success rate = 100% − 4% = 96%. Robot B is far better.

Rule: Before comparing anything, convert all values to the same format. Success rates, percentages, fractions - pick one and convert everything.

Students often confuse "probability of A given B" with "probability of B given A." These aren't the same thing.

"The probability that a person with the disease tests positive" is different from "the probability that a person who tests positive has the disease." When the question specifies a condition ("given that...", "considering only..."), check which group you're calculating within. The denominator changes depending on direction.

Stimulus:

Some red counters and ten black counters are put in a bag and drawn out randomly one at a time. They aren't replaced after they're drawn out. The first three counters drawn out are red. Peter predicts the next counter will also be red.

Question: "The number of red counters originally in the bag must have been at least 14 if Peter is more likely to be correct than incorrect."

Options:

• a) Yes, because with three reds drawn out the chance of the next counter being red is greater than one half.
• b) Yes, because the reds were drawn more often so there are better odds red will be drawn out next.
• c) Yes, because the reds were drawn more often so there are better odds black will be drawn out next.
• d) No, there can only have been 13 red counters in the bag originally.

Time check: The key calculation is just "11 vs 10 remaining after 3 draws." Once you see that 14 - 3 = 11 red vs 10 black, the answer is immediate. ~20 seconds.

Some questions ask about sequences of events:

"What is the probability of drawing two red counters in a row from a bag of 5 red and 3 blue?"

First draw: P(red) = 5/8
Second draw: P(red) = 4/7 (without replacement)
P(both red): 5/8 × 4/7 = 20/56 = 5/14

For "at least one" questions, use the complement:

"What is the probability of getting at least one head in two coin flips?"

P(no heads): 1/2 × 1/2 = 1/4
P(at least one head): 1 − 1/4 = 3/4

Some questions ask: "What's the minimum number of draws to guarantee an outcome?" These aren't really probability questions - they're worst-case questions.

Bag: 5 red, 3 blue, 2 green. "Minimum draws to guarantee 2 green counters?"

Worst case: you draw all the red and all the blue first - that's 5 + 3 = 8 non-green. Then the next 2 draws must be green.
Answer: 8 + 2 = 10 draws

Formula: Minimum to guarantee X of type T = (total non-T items) + X

When a PR question appears, spend 3 seconds classifying the subtype:

PR questions test several skills from the DM taxonomy:

• E1: Basic Probability Calculation - computing P = favourable / total for discrete outcomes
• E2: Misleading Framing / Unit Mismatch - catching deliberately mismatched formats (success rate vs. failure rate, percentages vs. fractions)
• E3: Combined / Joint Probability - multiplying probabilities for sequences, adjusting for without-replacement
• E4: Worst-Case / Guarantee (Pigeonhole Principle) - minimum draws to guarantee an outcome
• E5: Expected Value / Weighted Outcomes - multiplying probabilities by associated values

The comparison table handles E1 and E2 directly. Combined events (E3) need the multiplication rule. Pigeonhole questions (E4) need worst-case thinking, not probability at all.

1. Comparing mismatched units - 78% success vs. 4% failure looks like A > B until you convert. Always normalise first.
2. Forgetting "without replacement" - The denominator changes after each draw. 5/20 then 4/19, not 4/20.
3. Confusing "likely" with "guaranteed" - Probability questions ask what's most likely. Guarantee questions ask about worst-case certainty. Different logic entirely.
4. Confusing conditional directions - P(A given B) isn't the same as P(B given A). Check which group is your denominator.
5. Not re-reading the question - After calculating, check: did they ask for most likely or least likely? Highest or lowest? This reversal trap costs easy marks.

Next lesson: 2.3 Syllogisms