Question

Two brothers enter a race with five friends. The racers draw lots to determine their starting positions. What is the probability that the older brother will start in lane 1 with his brother beside him in lane 2?

Answer 1

Assumin the brothers are the first two to make draws the probability that they draw two specific lanes is given by
\[\frac{\left(\begin{matrix}2 \\ 2\end{matrix}\right)\left(\begin{matrix}5 \\ 0\end{matrix}\right)}{\left(\begin{matrix}7 \\ 2\end{matrix}\right)}=\frac{1}{21}\]
There are 2! permutations of the brothers' positions. So the probability of a specific permutation is
\[\frac{1}{21}\times \frac{1}{2}\]

Answer 2

Assuming*

Answer 3

thankss :) why do you multiply the 1/21 by 1/2 ?

Answer 4

The reason for multiplying 1/21 by 1/2 is as follows:
If the brothers are the first two competitors to make draws, the probability that one brother draws lane 1 and the other brother draws lane 2 is 1/21. But the required probability is that the older brother draws lane 1 and the younger brother draws lane 2.
There are 2 possible outcomes if one brother draws lane 1 and the other brother draws lane 2:
1. Younger brother draws lane 1 and older brother draws lane 2.
2. Older brother draws lane 1 and younger brother draws lane 2.
Only outcome #2 is allowed, therefore 1/21 must be divided by 2 to get the required probability.