Conditional probability

Given the weather: read the right branch

Question: the bus on a rainy day: There is a 30% chance of rain.

On rainy days, the bus has a 25% chance of being late.

Today, you know it is raining. What is the probability that the bus will be late?

Answer: 25%: We look only at rainy days. The question tells us that the bus is late on 25% of those days.

So P(late | rain) = 0.25. The symbol | means “given that”.

We know she won. Did she roll a 6?

The game: roll first, then spin: Mia rolls a fair six-sided die.

• If she rolls a 6, she uses spinner X. It has a 1/2 chance of winning.

• If she rolls 1, 2, 3, 4 or 5, she uses spinner Y. It has a 1/5 chance of winning.

Question: Mia has won: You know Mia won, but you did not see her die roll.

What is the probability that she rolled a 6?

Look only at the winning games. What fraction of those started with a 6?

Answer: 1/3, about 33.3%: Win after a 6: 1/6 × 1/2 = 1/12.

All wins: 1/12 + 2/12 = 3/12.

So P(rolled a 6 | won) = (1/12) ÷ (3/12) = 1/3.

The | means “given that”. We divide by the chance of all wins, because we already know she won.

Which games do we count?: We already know Mia won, so ignore every game she lost.

She could have won after rolling a 6, or after rolling 1–5. Count both kinds of win in the total.

Answer = chance of “roll a 6 AND win” ÷ chance of any win.

Here that is (1/12) ÷ (3/12) = 1/3. So one third of the winning games started with a 6.

Conditional probability from a table

Question: under-18 phone users: A phone shop surveys 200 customers, as shown in the table.

A customer is chosen at random. You are told they are under 18.

What is the probability that they use an iPhone?

Answer: 26/60 ≈ 0.433: Look only at the under-18 row: 60 customers, including 26 iPhone users.

So the probability is 26 ÷ 60 ≈ 0.433, or 43.3%.

Music and Drama: how many do both?

Question: 8 students are in the Music club only.

60% of the Music club also do Drama.

How many students do both Music and Drama?

The alarm rings — is there really smoke?

Picture a smoke alarm: A smoke alarm is meant to ring when there is smoke.

Sometimes it rings even though there is no smoke. That is a false alarm.

Information: 100 practice checks: A school carries out 100 practice checks on an alarm.

20 checks have smoke: the alarm rings in 18 of them.

80 checks have no smoke: the alarm still rings in 8 of them.

Question: One check is chosen at random.

You are told the alarm rang.

What is the probability that there was smoke?

The exam words, using this alarm: Positive means the alarm rings. Negative means it stays quiet.

False positive: it rings with no smoke — the 8 false alarms.

False negative: it stays quiet with smoke present — the 2 missed warnings.

Incorrect results: both mistakes together, so 8 + 2 = 10 out of 100, or 10%.