The probability tree of a multi-stage experiment

Probability, grade 11 (5 units) Lesson 9 of 16Learning

The probability tree of a multi-stage experiment

Stages, branches and paths

A probability tree describes an experiment of several stages. Each stage is a column of branches, and each branch carries its probability. A path – from the start to the end – is one outcome of the whole experiment.

A box holds 5 red balls and 3 white balls.
A ball is taken at random, not put back, and another ball is taken.

At the first stage there are 8 balls: red with probability $\displaystyle \frac{5}{8}$ , white with probability $\displaystyle \frac{3}{8}$ .
At the second stage 7 balls are left, and what they are depends on what came out first. After a red one, 4 red and 3 white are left; after a white one – 5 red and 2 white.

In the last column – the probability of each path: the product of the probabilities on its branches. Two red balls: $\displaystyle {\frac{5}{8}\cdot\frac{4}{7}}={\frac{20}{56}}={\frac{5}{14}}$ .

The probability of a path

The probability of a path is a product because of the definition of conditional probability. A branch at the second stage is a conditional probability: the probability that the second ball is red given that the first ball was red.

$\displaystyle R_1$ – the first ball is red; $\displaystyle R_2$ – the second ball is red.
By the definition $\displaystyle {P(R_2/R_1)}={\frac{P(R_1\cap R_2)}{P(R_1)}}$ , and multiplying by $\displaystyle P(R_1)$ :

\displaystyle P(R_1\cap R_2)

\displaystyle =

\displaystyle P(R_1)\cdot P(R_2/R_1)

P(R_1\cap R_2)=\frac{5}{8}\cdot\frac{4}{7}=\frac{5}{14}

In a path of three stages one more branch is multiplied in the same way: the probability of the third stage given the two stages before it.

An event of several paths

An event can be several paths. Different paths are different outcomes of the experiment – disjoint events – so their probabilities are added.

$\displaystyle A$ – “the two balls are of different colours”. The paths of $\displaystyle A$ are marked:

Red, then white: $\displaystyle {\frac{5}{8}\cdot\frac{3}{7}}={\frac{15}{56}}$ .
White, then red: $\displaystyle {\frac{3}{8}\cdot\frac{5}{7}}={\frac{15}{56}}$ .

P(A)=\frac{15}{56}+\frac{15}{56}=\frac{30}{56}=\frac{15}{28}

The probabilities of all the paths together make 1: $\displaystyle {\frac{20}{56}+\frac{15}{56}+\frac{15}{56}+\frac{6}{56}}={1}$ . This is a good way to check a tree.

Independent stages

When the stages are independent, the outcome of one stage does not change the probabilities of the next: $\displaystyle {P(B/A)}={P(B)}$ . So every fork of the same stage carries the same probabilities, and a path is simply the product of the stages’ probabilities.

A basketball player scores a free throw with probability $\displaystyle 0.8$ , whatever happens in her other throws.
She takes three throws.
The example below works out the probability that she scores exactly two of them. The event’s paths are marked:

Exactly two scores are three paths – the miss at the first, the second or the third throw. Each of them has two scores and one miss, so all three have the same probability:

3\cdot0.8\cdot0.8\cdot0.2=3\cdot0.128=0.384

“At least one” by the complement

The event “at least once” is many paths, but its complement is a single path: “never”. So work out the complement and take it away from 1.

What is the probability that the player misses at least one of the three throws?

The complement: she scores all three throws, one path: $\displaystyle {0.8^3}={0.512}$ .
At least one miss: $\displaystyle {1-0.512}={0.488}$ .

The long way would add up seven paths.

Exercise 1

In a certain town, in winter, the probability of rain on Sunday is $\displaystyle 0.3$ .
If it rains on a day, the probability of rain the next day too is $\displaystyle 0.6$ .
If a day is dry, the probability of rain the next day is $\displaystyle 0.2$ .
Complete the probabilities in the tree: the first stage – Sunday, the second stage – Monday.

What is the probability that it rains both on Sunday and on Monday?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power