Question 891951: If one person picks two letters from the alphabet at random (e.g. AB, CG, FO, ZZ [repetition is allowed]), and another person does likewise, what is the probability of the two people choosing the exact same letters?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
let's try a smaller set to see if we can come up with an algorithm that will give us the answer.
assume the alphabet only contained 3 letters (ABC)
the number of possible sets of 2 that you can get from this alphabet is equal to C(3,2) = 3.
Those 2 letter sets are:
AB
AC
BC
each person has the potential to get any of these 3 sets.
the total possible pairings from the 2 sets are therefore 3 * 3 = 9
those possible pairings are:
AB AB ***
AB AC
AB BC
AC AB
AC AC ***
AC BC
BC AB
BC AC
BC BC ***
if the first person chooses AB, then the probability that the second person will choose AB is 1/9.
if the first person chooses AC, then the probability that the second person will choose AC is 1/9.
if the first person chooses BC, then the probability that the second person will choose BC is 1/9.
the total probability appears to be 3/9 which is equal to 1/3.
it appears that the probability that they will match will be equal to C(3,2) / (C(3,2)^2 which is equal to 1 / C(3,2)
1 / (C(3,2) is equal to 1/3
let's see if this works with 4.
you have 4 letters in the alphabet.
they are ABCD
you are picking 2 out of the 4.
the possible combinations are C(4,2) = (4*3)/(1*2) = 6
those possibilities are:
AB
AC
AD
BC
BD
CD
that' 6 possible sets of 2 for each person.
the possible pairings should be 6 * 6 = 36
those possible pairings are:
AB AB *****
AB AC
AB AD
AB BC
AB BD
AB CD
AC AB
AC AC *****
AC AD
AC BC
AC BD
AC CD
AD AB
AD AC
AD AD *****
AD BC
AD BD
AD CD
BC AB
BC AC
BC AD
BC BC *****
BC BD
BC CD
BD AB
BD AC
BD AD
BD BC
BD BD *****
BD CD
CD AB
CD AC
CD AD
CD BC
CD BD
CD CD *****
out of the 36 possible pairings, there are 6 pairings that match.\
the probability of getting a match is therefore 6 / 36 = 1/6
it looks like the formula is holding.
the formula is 1 / C(n,x)
in this case the formula is 1 / C(4,2) which is equal to 1/6.
in the case of 3 letter alphabet, the formula was 1 / C(3,2) = 1/3,.
both cases held true, so we can reasonably assume that the same formula will work with larger numbers.
the formula of 1 / C(n,x), applied to a 26 letter alphabet, yields 1 / C(26,2) = 1 / 325.
I believe that's your answer.
It's hard to prove directly, but it does appear to be good after we tried it on 2 smaller sets of letters.
we can also try to look at it another way.
assume you roll 2 dice that have 6 sides each.
the probability of getting a 1 on the first die is 1/6
the probability of getting a 1 on the second die is 1/6
the probability of both die getting a 1 is 1/6 * 1/6 = 1/36.
the probability of getting a 2 and a 2 is also equal to 1/36
same for 3,4,5,6.
the total probability of getting a match on 1 or 2 or 3 or 4 or 5 or 6 is the sum of the individual probabilities is equal to 1/36 * 6 = 6/36 = 1/6.
now assume the die have 325 sides.
the same problem becomes 1/325^2 + 1/a325^2 + ....... + 1/325^2 = 325 / 325^2 = 1/325.
we get the same answer.
that appears to be your probability based on two sets of reasoning.
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