Question 891951



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.