Probability, and why your intuition about it is wrong
The maths is simple. Your instincts are the problem.
Probability has a reputation for being difficult. The calculations mostly are not - at this level it is counting followed by a division. What makes it hard is that human intuition about chance is not merely imprecise, it is wrong in specific, repeatable ways.
The basic definition
For outcomes that are all equally likely, the probability of an event is the number of outcomes you want divided by the total number of outcomes.
- One die, rolling a 4: one wanted outcome out of six, so 1/6
- One die, rolling an even number: three out of six, so 1/2
- A card being a heart: thirteen out of fifty-two, so 1/4
Everything harder is still this, applied to a set of outcomes that takes more work to count - which is why combinatorics comes first.
And versus or
Two rules cover most questions, and mixing them up is the most common error in the topic.
- AND means multiply (for independent events). Two heads in a row: 1/2 x 1/2 = 1/4.
- OR means add (for events that cannot both happen). Rolling a 1 or a 2: 1/6 + 1/6 = 1/3.
The word independent is doing real work in the first rule. Two events are independent when one happening does not change the chance of the other. Coin flips are independent. Drawing two cards without replacing the first is not - the second draw has 51 cards to choose from, and the deck has changed.
Whenever a question says 'at least one', calculate the chance of none and subtract from 1. It converts a long sum into one multiplication, and it works every time.
Where intuition fails
The gambler's fallacy
A coin lands heads six times running. People feel tails is now 'due'. It is not. The coin has no memory and the chance is still exactly 1/2. Nothing about the previous flips reaches the next one.
The reverse error is just as common: concluding the coin is biased toward heads. Six heads in a row happens by chance about once in every 64 sequences of six flips, which is unremarkable.
Confusing rare with impossible
With enough trials, unlikely things become near-certain. A one-in-a-thousand event, given ten thousand chances, will almost certainly happen several times. Lotteries are won every week by someone, even though each individual ticket is hopeless.
Ignoring the base rate
This one is genuinely counter-intuitive and worth walking through. Suppose a test for a condition is 99% accurate, and the condition affects one person in ten thousand. You test positive. What is the chance you have it?
Most people say around 99%. Take a million people. About 100 have the condition, and the test catches 99 of them. The other 999,900 do not have it, and 1% of those - about 9,999 people - test positive anyway. So roughly 10,098 positive results, of which 99 are real. That is under 1%.
The test is not bad. The condition is simply so rare that false positives outnumber true ones. This is why doctors retest, and why a single positive result on a screening test for a rare thing is not a diagnosis.
Back to your own multiple-choice papers. Five options and no penalty for wrong answers means a blind guess is worth 1/5. Eliminate two options and it becomes 1/3. Over forty questions that difference is several marks, which is why elimination pays even when you cannot finish the reasoning.
Expected value
Multiply each outcome by its probability and add them up. It tells you the average result over many repetitions.
A ticket costing 1 pound with a 1-in-10 chance of returning 8 pounds has expected value 0.1 x 8 = 0.80, against a cost of 1. Every play loses 20p on average. You might win. Over a thousand plays you will be down about 200 pounds, and that is the number that matters for any decision you make repeatedly.