How a Condition Changes a Probability

How a Condition Changes a Probability — Moosa Academy

Probability questions reward careful reading. Every extra phrase is doing work: it can raise the probability, lower it, or wipe it out entirely. The reason is always the same — a condition shrinks the set of outcomes still in play.

Concept The set-up: four cars in a draw

Four cars, one of each kind, all equally likely to be drawn. With nothing else said, each individual car has probability

\tfrac{1}{4}

The four cars form the sample space — the complete list of things that could happen. Every probability in this lesson is a count of favourable cars divided by the size of that space.

Example No condition at all

P(\text{Toyota}) = \dfrac{2}{4} = \dfrac{1}{2}

— two of the four cars are Toyotas

P(\text{small Toyota}) = \dfrac{1}{4}

— only one car fits

Half the cars are Toyotas, half are Fords, and each specific car is 25%. This is the baseline that the conditions will change.

Example A condition that raises the probability

What is the probability of a small Toyota, given that the car is a Toyota?

Step 1 — the condition rules out both Fords

Step 2 — the sample space shrinks from four cars to two: small Toyota and large Toyota

Step 3 — one of those two is favourable

P(A \mid B) = \frac{1}{2}

⟹ up from

\tfrac{1}{4}

\tfrac{1}{2}

Nothing about the car changed — only what is known about it. Knowing it is a Toyota removes half the competition, so the same single car now takes a larger share.

Example A condition that eliminates it

What is the probability of a large Toyota, given that the car is a Ford?

Step 1 — the condition says the car is definitely a Ford

Step 2 — a car cannot be a Ford and a Toyota at the same time

Step 3 — no favourable outcomes remain

P(A \mid B) = 0

⟹ the condition made it impossible

These two events are mutually exclusive: they cannot both happen. When the condition is one of them, the other drops to zero.

Note The three effects side by side

P(\text{small Toyota}) = \tfrac{1}{4}

— no condition, the baseline

P(\text{small Toyota} \mid \text{Toyota}) = \tfrac{1}{2}

— raised

P(\text{large Toyota} \mid \text{Ford}) = 0

— eliminated

A condition can also lower a probability without destroying it — that happens when it narrows the sample space but throws away favourable outcomes at the same time. In every case the rule is one and the same: a condition restricts the sample space, and the probability is recalculated inside whatever remains.

Summary

A condition restricts the sample space to the outcomes that satisfy it.
With four equally likely cars, each specific car has probability 1/4.
Given that the car is a Toyota, a small Toyota rises to 1/2 — two cars remain, one is favourable.
Given that the car is a Ford, a large Toyota falls to 0 — the events are mutually exclusive.
Read every phrase in a probability question: each one may change the answer.