A sweet paradox
A frined of mine likes sweets. To be precise, he likes sweets of different tastes. Once he bought four sweets, two strawberry and two lemon.
- How many sweets will you eat? I asked.
- Just two. And I hope they are of different taste.
- Well, they may be, you know. If you are lucky.
- What do you mean?
- Well, if you pick two sweets at random there are 4 possible ways of doing it, and 2 of these have different tastes.
- So you are saying that my chances are 1/2?
- Exactly.
- I have found a better way!
- Really?
- I eat one sweet, it doesn’t matter which.
- And then?
- Of the three remaining sweets 2 have a different taste than the first so my chances are 2/3 for getting two sweets of different taste.
- You must be a genius. I thought 1/2 was all you could hope for.
Problem source: Activa sus neurona by laberintos e infinitos.
April 30th, 2009 at 2:23 am
Not so. Here are two ways of finding the correct chance. Label the strawberry sweets s1, s2 and the lemon ones l1, l2.
1. Counting sequences
Count the possible sequences in which you could pick 2 sweets:
s1 s2
s1 l1 #
s1 l2 #
s2 s1
s2 l1 #
s2 l2 #
l1 s1 #
l1 s2 #
l1 l2
l2 s1 #
l2 s2 #
l2 l1
There are 12 sequences, all equally likely, and 8 of them (labelled #) yield 2 sweets of different flavours. Thus the chance that this happens is 2/3.
2. Counting combinations
Here we count merely the possibilities for which 2 sweets were picked, ignoring the order in which the sweets were drawn.
s1 s2
s1 l1#
s1 l2#
s2 l1#
s2 l2#
l1 l2
There are 6 combinations, all equally likely, and 4 of them (labelled #) yield 2 sweets of different flavours. Thus the chance that this happens is 2/3.