GB-18-04-001 · Two draws without replacement

A box contains 8 black pens and 20 blue pens. Two pens are taken at random without replacement. Find the probability that they have different colours.

Probability Trees (Without Replacement) Booklet p. 60

Method

1) Note item is NOT replaced so second denominator decreases by 1

2) Multiply along branches for P(B and L) and P(L and B)

3) Add the two probabilities

Worked Example (GB-18-04-001)

A = black; B = blue. Do not replace the first item: the second total is one smaller, and the first colour count decreases by one.

First draw	P(first)	Second draw	P(second \| first)	Path probability
A	$\frac{2}{7}$	A	$\frac{7}{27}$	$\frac{2}{7}\times\frac{7}{27}$
A	$\frac{2}{7}$	B	$\frac{20}{27}$	$\frac{2}{7}\times\frac{20}{27}$
B	$\frac{5}{7}$	A	$\frac{8}{27}$	$\frac{5}{7}\times\frac{8}{27}$
B	$\frac{5}{7}$	B	$\frac{19}{27}$	$\frac{5}{7}\times\frac{19}{27}$

Multiply along each path, then add the two different-colour paths:

P(AB)+P(BA)=\frac{2}{7}\times\frac{20}{27}+\frac{5}{7}\times\frac{8}{27}=\frac{80}{189}

Ans =

\frac{80}{189}