Comic Relief is nearly upon us once more; the extraordinary British charity event that results in the biennial reinvention of the red nose. Every two years you can either fork out for a new ruddy conk or throw caution to the wind and reach down the back of your sofa to find the one from last time. And some peanuts. And something that used to be less slimy.
This year, there are nine of the handsome little hooters to choose from.
The catch is, however, that you can't choose which one you get because they come in surprise opaque packages. This provokes the following question: If you really wanted to get hold of Snorbit (I don't know, maybe cyloptic aliens are your thing) then how much would you expect to donate before you found one?
At this stage, the traditional unit price of £1 makes the mathematics a little bit more straightforward; the red nose has been brilliantly blowing raspberries at inflation for years!
You might, with a probability of 1/9, get very lucky and find Snorbit at your first attempt. We are of course assuming that each nose is an equally likely pick (pun definitely intended).
If you get any other schnozzle (with probability 8/9) you might get lucky on the second go (probability 1/9) and pay £2.
You could end up donating £3, in which case you have been unlucky twice (8/9 squared) and third-time lucky (1/9).
This pattern continues forever (or at least until all the non-Snorbit noses run out, but let's not be pedantic) and we can find our expected donation by summing the following series:
A cunning bit of rearranging has left us with an infinite geometric series. Using the relevant formula known to all sixth-form mathematicians, it can be easily seen that the sum of this is
Suppose however that you were more ambitious. How much would you expect to have to donate in order to collect at least one of every nose?
Well you wouldn't have to wait long until you collected your first nose; it is inevitable that the donation of your first pound would grant you this.
From then onwards, however, it isn't quite straight forward. We've already seen that you have to donate £9 on average to pick a particular nose. This is equivalent to having eight noses already and requiring the ninth.
What if you were on seven noses? How much would you expect to have to donate to get your eighth? There are two noses left so you'll pick one of them with 2/9 probability on any particular go. However, in 7 out of every 9 turns you'll pick one you already have. Therefore the amount you can expect to donate in upgrading from seven noses to eight noses is
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This year, there are nine of the handsome little hooters to choose from.
 |
| Stripey, Astrosnort, Supernose, Snotty Professor, Nosebot, Snorbit, Snortel, Karate Konk, and Snout Dracula |
The catch is, however, that you can't choose which one you get because they come in surprise opaque packages. This provokes the following question: If you really wanted to get hold of Snorbit (I don't know, maybe cyloptic aliens are your thing) then how much would you expect to donate before you found one?
At this stage, the traditional unit price of £1 makes the mathematics a little bit more straightforward; the red nose has been brilliantly blowing raspberries at inflation for years!
You might, with a probability of 1/9, get very lucky and find Snorbit at your first attempt. We are of course assuming that each nose is an equally likely pick (pun definitely intended).
If you get any other schnozzle (with probability 8/9) you might get lucky on the second go (probability 1/9) and pay £2.
You could end up donating £3, in which case you have been unlucky twice (8/9 squared) and third-time lucky (1/9).
This pattern continues forever (or at least until all the non-Snorbit noses run out, but let's not be pedantic) and we can find our expected donation by summing the following series:
A cunning bit of rearranging has left us with an infinite geometric series. Using the relevant formula known to all sixth-form mathematicians, it can be easily seen that the sum of this is
So £9 is the mean donation required to pick Snorbit. Of course, you could be luckier or unluckier. In actual fact, you are more likely to pick Snorbit on the first attempt than any other subsequent attempt as the graph below demonstrates:
Suppose however that you were more ambitious. How much would you expect to have to donate in order to collect at least one of every nose?
Well you wouldn't have to wait long until you collected your first nose; it is inevitable that the donation of your first pound would grant you this.
From then onwards, however, it isn't quite straight forward. We've already seen that you have to donate £9 on average to pick a particular nose. This is equivalent to having eight noses already and requiring the ninth.
What if you were on seven noses? How much would you expect to have to donate to get your eighth? There are two noses left so you'll pick one of them with 2/9 probability on any particular go. However, in 7 out of every 9 turns you'll pick one you already have. Therefore the amount you can expect to donate in upgrading from seven noses to eight noses is
This looks very similar to the type of series we had before and, sure enough, a similar trick shows us that it converges to £4.50.
Working backwards, the same method shows that the amount I'd expect to donate upgrading from six different noses to seven different noses is £3.
I can add up the expected cost to get one nose, then the upgrade to two different noses, then the upgrade from two to three different noses, and so on. This gives the following sum:
So, if you manage to collect all nine noses for £25 or less, you can consider yourself lucky. Of course you could spend £100 and still not have the whole set of snouts but the probability of this happening is 0.000069 and at least, in this case, you can have a good chuckle at what some consider to be the funniest number.
from matheminutes http://ift.tt/1EgkB7D
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