Example 1: Board of Studies Syllabus, p33.

5 boy's names and 6 girl's names are in a hat. Find the probability that in 2 draws a boy's name and a girl's name are drawn. (No replacement of names after a draw.)

ANSWER

There is no distinction amongst the names other that they are $b$ or $g$, $S=\{b,b,b,b,b, g,g,g,g,g,g \}$.

In a two stage problem such as this, we have to evaluate the effects of both draws because upon removal of the first name, the sample space $S$ is reduced by 1.

Let $E_1$ be the event that the draw goes $\{b,g\}$ and $E_2$ be the event the draw results are $\{g,b\}$, and $E$ be the event of either $E_1$ or $E_2$,

$$\begin{array}{lclll} E_1 & = & b_1 \cap g_2 & & \\ E_2 & = ... ...11}} \times {5 \over {10}}& = {6 \over {11}} \\ \end{array}$$

It is not always necessary to resort to strict formalism and the syllabus recommends a tree diagram (Figure 5) to aportion the correct probabilities.

Figure 5: Probability Tree Example 1

The tree is a convenient way of implementing the steps of the formal maths and it also requires that you take care to define your terms and apply the probability results (1) - (5) correctly. And of course whatever strategy you take, the setting out must be clear so that the examiner can follow your reasoning.