### Knock-out

Before a knockout tournament with 2^n players I pick two players. What is the probability that they have to play against each other at some point in the tournament?

### Squash

If the score is 8-8 do I have more chance of winning if the winner is the first to reach 9 points or the first to reach 10 points?

### Snooker

A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

# Heavy Hydrocarbons

##### Stage: 4 and 5 Challenge Level:

This problem consists of a series of questions concerning hydrocarbons made from the stable isotopes of hydrogen and carbon. Assume that the probability of the occurrence of unstable isotopes is effectively zero. Assume also that the mass of an isotope is equal to the mass number.

Carbon is found in two naturally occurring stable isotopic forms: Carbon-12, $^{12}$ C, accounts for $98.9\%$ of naturally occurring carbon, and Carbon-13, $^{13}$ C, accounts for $1.1\%$ of naturally occurring carbon.

Hydrogen also has two stable isotopes: $^1$H accounts for slightly more than $99.985\%$ of naturally occurring hydrogen. $^2$H, called deuterium, accounts for about $0.015\%$ of naturally occurring hydrogen.

What possibilities are there for the molecular mass of a stable molecule of methane CH$_4$? Which three are the most common? What is the probability that a randomly encountered methane molecule will be of each of these masses?

What would be the three most likely possibilities for the molecular masses of the next two alkanes: ethane, C$_2$H$_6$, and propane C$_3$H$_8$?

Am I likely to find a molecule of butane C$_4$H$_{10}$ of molecular mass $72$ in a litre of butane? (Avogadro's constant is approximately $6.02\times 10^{23}$).

Are there any alkanes for which the most commonly found molecule is not the one made entirely from $^{12}$C and $^1$H?

NOTES AND BACKGROUND

Mass spectrometry allows chemists to determine the molecular masses of compounds. The frequency of occurrence of common isotopes of elements is an important factor in understanding such as analysis. Mass spectrometry is of great industrial and scientific importance, and in practice, the results of an analysis are determined by comparison with a database of experimental spectrometry information.

For information on alkanes and hydrocarbons, see http://en.wikipedia.org/wiki/Hydrocarbon