I'm looking for an explanation of a probability value.
The UK has a lottery game called Thunderball. The player chooses 5 Main Numbers from 1 to 39 and also 1 Thunderball from 1 to 14. The Thunderball machine does the same and if all of your 6 numbers match then you win the jackpot.
I understand that the odds of winning the jackpot are 1 in 8,060,598.
What I don't understand is why the odds of matching ONLY the Thunderball are 1 in 29?
Surely it should be 1 in 14?
For more info on the Thunderball game, please visit Wikipedia at: https://en.wikipedia.org/wiki/National_Lottery_(United_Kingdom)#Thunderball
Thank you for your help.
Best Answer
The odds of winning the jackpot is quite easy indeed, and we have
$$ 8060598 = \binom{39}{5}\times 14$$
and indeed, you have one chance over 14 to get the thunderball number correct. However here they present the odds of winning only the thunderball. Therefore you need to deduce the odds of guessing the thunderball and one or more normal numbers.
To compute the result, you need to have a look at how many combination of the 5 main numbers can be dealt by the machine with not as single one in common with you : hence $\binom{34}{5}$. Therefore the proportion of these combinations is $$ \binom{34}{5}/\binom{39}{5}\approx 0.48\%$$
And $$14\times\frac{\binom{39}{5}}{\binom{34}{5}}\approx 28.97$$