More general trees

It is straight forward to generalise the tree diagram in several ways without changing its properties (1-3). One possibility is to add more generations. Consider a random experiment with three events $A$ , $B$ , and $C$ . We can form the following tree:

Here,

p_1=p(A) \text{ and }p_2=p(A^\prime)

and all other probabilities are conditional probabilities. E.g. $p_3$ is the probability that $B$ occurs, given that $A$ has occurred

p_3=p(B\vert A)

and $p_7$ is the probability that $C$ occurs given that $A$ and $B$ have occurred

p_7=p(C|A \cap B)

The product of the branch probabilities $p_1\cdot p_3\cdot p_7$ is then the probability that $A$ , $B$ , and $C$ occurred simultaneously:

p(A\cap B \cap C)=p_1\cdot p_3\cdot p_7

Another possibility to extend the tree is to use more than two branches per node. So, consider three events $A_1, A_2$ and $A_3$ which form a partition of the sample space $S$ , and another three events $B_1, B_2$ and $B_3$ which also form a partition of $S$ . Recall that a partition divides the sample space. Thus, whenever the experiment is performed, exactly one of the events $A_1, A_2$ and $A_3$ will occur, and also exactly one of the events $B_1, B_2$ , and $B_3$ will occur. Thus

p(A_1)+p(A_2)+p(A_3)=1

and

p(B_1)+p(B_2)+p(B_3)=1

(as was the case for the tree with two branches per node, where the partitions were $A$ and $A^\prime$ , and also $B$ and $B^\prime$ ). We draw the tree as follows:

And here are some exercises ...

Exercise 1

Q1

In a town, $30\%$ of the population reads newspaper A, of these $50\%$ read newspaper B, and of these $10\%$ read newspaper C. A person is selected at random from the town. What is the probability that the person reads all three newspapers?

Q2

A box contains 4 blue balls and 5 red balls. Four balls are selected at random without replacement. What is the probability for selecting

exactly $4$ blue balls.
exactly $3$ blue balls.
no blue ball.
a blue ball given that the previous three balls were red.

Q3

A box contains 4 blue balls and 5 red balls. Four balls are selected at random with replacement. What is the probability for selecting

exactly $4$ blue balls.
exactly $3$ blue balls.
a blue ball given that the previous three balls were red.

Q4

A small brewery has three bottling machines. Machine A fills $40\%$ of all bottles, machine B and C fill $30\%$ each. $5\%$ of the bottles filled by A, $4\%$ of the bottles filled by B, and $3\%$ of the bottles filled by C are rejected for some reason. If a bottle is filled by A or B, what is the probability that it is rejected?

Q5

A box contains 2 blue balls, 3 red balls and 4 yellow balls. Two balls are selected at random without replacement. What is the probability that

the selected balls have different colour?
the first ball is yellow given that the last ball is blue?

Solution

More general trees

Q1

Q2

Q3

Q4

Q5

A1

A2

A3

A4

A5