Determining Probabilities Systematically

Rather than using Contingency Tables, since these can break once we have more than 2 events, we use a system of equations. The processes is to:

Determine probabilities $P (E_{i})$ for all simple events where all are $\geq 0$ and $\sum P (E_{i}) = 1$
Find $P (A)$ which is a compound event by adding the relevant $P (E_{i})$ 's for each $E_{i} \in A$ .

A Train Car Example

A train has 5 cars. Suppose a commuter is twice as likely to select the middle car $# 3$ as compared to its adjacent car $# 2, # 4$ . Further, they are twice as likely to select either end car $# 1, # 5$ compared to the adjacent car. Thus:

p_{3} = 2 p_{2} = 2 p_{4}; p_{2} = 2 p_{1} = 2 p_{5} = p_{4}

Where each $p_{i} = P (car i selected) = P (E_{i})$ . Thus:

1 = \sum P (E_{i}) = p_{1} + 2 p_{1} + 4 p_{1} + 2 p_{1} + p_{1} = 10 p_{1} \Rightarrow p_{1} = .1

Thus $p_{2} = .2 = p_{4}, p_{3} = .4$ . The probability of choosing none of the end cars is $p_{2} + p_{3} + p_{4} = .8$