Two stages: the two-way table and the tree

In a two-way table each person is counted once, by two questions together. When a person is chosen at random from the whole table, divide by the grand total.

One winter, a school recorded who had a flu jab and who caught flu:

A pupil is chosen at random from all 800 pupils.

• The probability that they caught flu: $\displaystyle \frac{93}{800}$.
• The probability that they had the jab and caught flu: $\displaystyle \frac{18}{800}$.

When the question is about part of the population – “of those who had the jab” – divide by the row’s total, not by the table’s.

• Of the 300 who had the jab, 18 caught flu: $\displaystyle {\frac{18}{300}}={0.06}={6\%}$.
• Of the 500 with no jab, 75 caught flu: $\displaystyle {\frac{75}{500}}={0.15}={15\%}$.

In this school, those with no jab caught flu at a rate 2.5 times that of those who had it.
This comparison cannot be made from the numbers themselves, 18 and 75: the groups differ in size.

Even a good test is sometimes wrong. When a disease is rare, most of the people whose test comes out positive are healthy.

A disease affects 1% of the population.
The test finds it in 90% of those who have it.
In 5% of healthy people the test comes out positive by mistake.

Here is a table of 10,000 people tested:

• Ill: 1% of 10,000, that is 100. The test is positive for 90% of them: 90.
• Healthy: 9,900. The test is positive for 5% of them: $\displaystyle {0.05\cdot9900}={495}$.

Of the 585 people with a positive test, only 90 are ill:

That is why a positive test is followed by another test before anyone is told they are ill.

When the second stage does not depend on the outcome of the first, draw a tree diagram – as in grade 9. A path’s probability is the product of the probabilities on its branches:

There are two traffic lights on the way to school.
The probability of reaching the first one on green is $\displaystyle \frac{2}{5}$, and the second – $\displaystyle \frac{1}{2}$.
The path “both lights green” is marked on the tree:

• Both lights green: $\displaystyle {\frac{2}{5}\cdot\frac{1}{2}}={\frac{1}{5}}$.
• At least one red light – the complement: $\displaystyle {1-\frac{1}{5}}={\frac{4}{5}}$.

A survey asked 400 drivers whether they use a phone while driving, and whether they had been in an accident in the last three years:

A driver is chosen at random from all the drivers in the survey. What is the probability that they had an accident?

What is the probability that a driver who uses a phone had an accident?

What is the probability that a driver who does not use a phone had an accident?

Does the survey prove that using a phone causes accidents? Explain.

A footballer scores a penalty with probability $\displaystyle \frac{4}{5}$.
She takes two penalties, and each kick does not depend on the one before.

Write the probabilities on the branches and each path’s probability.
Mark the paths of the event “she scores at least once”, and find its probability.

The probability that a newborn baby is a boy is about $\displaystyle \frac{51}{100}$.
A family has two children.

Write the probabilities on the branches and each path’s probability.

What is the probability that both children are girls?

What is the probability that the family has at least one boy?

The class committee has 6 girls and 4 boys.
Two of them are drawn, one after the other, to represent the class on the school council.

Write the probabilities on the branches and each path’s probability.
What is the probability that both representatives are girls?

2,000 people were tested, and 100 of them have a certain disease.
The test came out positive for 80 of those who are ill.
For 190 of the healthy people the test came out positive by mistake.

What is the probability that a person whose test is positive is really ill?

Ronnie got a positive result. What would you tell him? Explain with the answer to part a.