Probabilistic Reasoning

Probability questions give you a scenario with numerical data - counters in a bag, success rates of machines, prize-draw odds, transport reliability - and ask which option is most/least likely, or whether a probability claim is valid. Format: 4 options, 1 mark.

These make up ~4-5 questions (~11% of DM). Target time: 15-30 seconds for simple questions; 45-60 seconds for combined events.

Target time and triage

Probability is one of the best mark-per-minute trades in DM - the maths is GCSE-level, the patterns are repetitive, and only two rules cover the majority. Do these on the first pass.

The catch: probability questions are placed last in the DM exam order, so candidates who work front-to-back run out of time and miss them. The cure is triage: skip LP and complex Venn early to keep time for the last few probabilities.

Two ideas account for most UCAT probability questions:

Rule 1 - The complement rule

   P(at least one) = 1 − P(none)
   P(not A)         = 1 − P(A)

Whenever a question asks "probability of at least one X" - STOP. Don't add up "exactly one + exactly two + exactly three…". Compute P(no X) and subtract from 1. It's almost always faster.

Rule 2 - Translation / same-terms rule

Comparison questions deliberately phrase the two options in different units. One is given as a success rate; the other as a failure rate. One as a probability; the other as a frequency ("17 out of 20"). Before you compare, convert both to the same term.

  Crossfield: 0.15 probability of running LATE.
  Riverside:  on time 17 out of 20 journeys.

  Convert Riverside to P(late) = 3/20 = 0.15.
  Equal. (The trick is to spot this isn't 0.85 vs 0.15.)

These two rules drive both worked examples below.

Tanya enters two separate prize draws. The probability of winning the first is 0.3. The probability of winning the second is 0.4.

Question: Is Tanya more likely than not to win at least one prize?

Naive approach. Compute the three winning scenarios and add:

• P(wins first only) = 0.3 × 0.6 = 0.18
• P(wins second only) = 0.7 × 0.4 = 0.28
• P(wins both) = 0.3 × 0.4 = 0.12
• Sum = 0.58. "More likely than not" threshold is 0.5. Yes.

Three multiplications, one addition, one comparison. Easy to slip.

Complement approach.

• P(wins at least one) = 1 − P(wins none)
• P(wins none) = 0.7 × 0.6 = 0.42
• P(wins at least one) = 1 − 0.42 = 0.58. More than half.

One multiplication, one subtraction, one comparison. Same answer, half the work.

*Reach for the complement whenever you see "at least one," "at least two," or "probability of getting any…"*** - and equally whenever you'd otherwise face a long sum of mutually exclusive cases.

Crossfield coaches and Riverside buses both operate the same route.
- Crossfield has a 0.15 probability of running late.
- Riverside services run on time on 17 out of every 20 journeys.
- Crossfield offers a refund on three quarters of its delayed services.
- Riverside has a 25% probability of not offering a refund when its service is delayed.

Question: Considering only punctuality and likelihood of receiving a refund when delayed, is Crossfield the better choice?

Build a comparison table. Two attributes, two options.

Same terms now. P(late) is identical. P(refund when delayed) is identical. Crossfield is not better - they're equivalent. Answer: No.

The translation step is the whole technique. Once both numbers are in the same units, the answer falls out.

You will not need conditional probability formulas, Bayes' theorem, or expected value tables. This is the entire toolkit.

The single most common PR trick: numbers given in deliberately mismatched formats.

Rule: before comparing anything, convert all values to the same unit.

P(A given B) ≠ P(B given A). The denominator changes depending on which group you're conditioning on.

"The probability that a person with the disease tests positive" (denominator = people with the disease)

≠

"The probability that a person who tests positive has the disease" (denominator = people who tested positive)

When you see "given that", "considering only", "out of all the…" - check which group sits in the denominator before computing.

Sequences of events use multiplication, with the without-replacement adjustment if relevant.

Bag of 5 red and 3 blue. Two draws without replacement. P(two reds)?

First draw: P(red) = 5/8.
Second draw, given first was red: P(red) = 4/7.
P(both red) = 5/8 × 4/7 = 20/56 = 5/14.

For "at least one" questions, always use the complement instead of summing.

Some questions ask "minimum number of draws to guarantee X." These aren't probability questions at all - they're worst-case questions.

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

Worst case: you draw every red and every blue first → 5 + 3 = 8 non-green.
The next 2 draws must be green.
Answer: 8 + 2 = 10 draws.

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

1. Comparing mismatched units. Always translate first.
2. Summing cases instead of using the complement. The moment you see "at least one," reach for 1 − P(none).
3. Forgetting without-replacement. After each draw, both numerator and denominator drop by 1.
4. Confusing "likely" with "guaranteed". Probability questions are about likelihood; pigeonhole questions are about worst-case certainty. Different logic.
5. Mixing up conditional direction. P(positive | disease) ≠ P(disease | positive).
6. Not re-reading the question. "Most likely" vs "least likely," "more than half" vs "less than half." Reversal traps cost easy marks.

Probability questions test five skills:

• E1: Basic Probability Calculation - computing P = favourable / total for discrete outcomes.
• E2: Misleading Framing / Unit Mismatch - catching deliberately mismatched formats (success vs failure, fraction vs probability).
• E3: Combined / Joint Probability - multiplying 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 their associated values.

The two key rules (complement, translation) cover E1, E2, and most of E3. E4 is pigeonhole. E5 is rare and follows from E1 + multiplication.