High School Where Did I Go Wrong in Calculating the Probability?

[little neutrino]

A code is generated consisting of 3 letters followed by 2 digits. Each of the 3 letters generated is equally likely to be any of the 26 letters of the alphabet. Each of the 2 digits generated is equally likely to be any of the nine digits 1-9. Find the probability that a randomly chosen code has exactly 2 letters the same or 2 digits the same, but not both.

Attempted solution:

P(exactly 2 digits same) = (26³*9)/(26³*9²) = 1/9
P(exactly 2 letters same) = (26*25*3*9²)/(26³*9²) = 75/676
P(2 digits same AND 2 letters same) = (26*25*3*9)/(26³*9²) = 25/2028
Required probability = 1/9 + 75/676 - 25/2028 = 0.210

But the correct answer is 0.197. Can I ask where did I go wrong? I can't figure it out...
Thanks!

 

[stevendaryl]

A code is generated consisting of 3 letters followed by 2 digits. Each of the 3 letters generated is equally likely to be any of the 26 letters of the alphabet. Each of the 2 digits generated is equally likely to be any of the nine digits 1-9. Find the probability that a randomly chosen code has exactly 2 letters the same or 2 digits the same, but not both.

Attempted solution:

P(exactly 2 digits same) = (26³*9)/(26³*9²) = 1/9
P(exactly 2 letters same) = (26*25*3*9²)/(26³*9²) = 75/676
P(2 digits same AND 2 letters same) = (26*25*3*9)/(26³*9²) = 25/2028
Required probability = 1/9 + 75/676 - 25/2028 = 0.210

But the correct answer is 0.197. Can I ask where did I go wrong? I can't figure it out...
Thanks!

Your calculation for 2 letters is the probability that at LEAST 2 letters is the same. You have to subtract off the probability that all 3 letters are the same to get the probability that exactly 2 letters are the same.

 

[cpscdave]

One other possible mistake. Exactly how many ways are there to have 2 items the same in a set of 3?

 

[little neutrino]

Your calculation for 2 letters is the probability that at LEAST 2 letters is the same. You have to subtract off the probability that all 3 letters are the same to get the probability that exactly 2 letters are the same.

But why is that so?
__ __ __ __ __
For two letters to be the same == 26 choices
For the last letter, since it can't be the same as the two letters that are the same == 26 - 1 = 25 choices
No. of ways to arrange 3 letters, when 2 are the same == 3!/2! = 3
Thus no. of ways == 26 * 25 * 3 * 9²

Why did I calculate the probability that at least 2 letters are the same? Could you elaborate? Sorry kind of confused.

 

[cpscdave]

No. of ways to arrange 3 letters, when 2 are the same == 3!/2! = 3

Yup I'm dumb sorry :D

 

[stevendaryl]

Hmm. My criticism was mistaken, but here's a suggestion: Instead of computing probabilities for the various possibilities, try just computing

N_L = the number of ways to have exactly 2 letters the same, and both digits different.
N_D = the number of ways to have 3 or 0 letters the same, and both digits the same.

Then the probability is \frac{N_L + N_D}{26^3 \cdot 9^3}

When I did it that way, I got the expected answer.

 

[stevendaryl]

Okay, I figured out what you did wrong:

You used P(A \vee B) = P(A) + P(B) - P(A \wedge B) where A = "exactly 2 letters the same" and B = "both digits the same". But they aren't asking for A \vee B, they're asking for A\ xor\ B (xor = exclusive or). The formula for that is:

P(A\ xor\ B) = P(A) + P(B) - 2P(A \wedge B)