# Probability of events having to deal with a string permutation question

I am studying for a discrete math exam tomorrow and this is one of the review questions. I am having trouble answering the question as of now. If you could provide guidance on how to solve one or more of the following parts of the question, I would be really appreciative!! Let me know if you need any other information!

Assume that a string of length 9 is a random permutation of letters {a, b, c, d, e, f, g, h, i}.

(a) What is the probability of event A: letter g comes before a ?

(b) What is the probability of event B: letter g comes after b and d ?

(c) What is the probability of event C: letters e and f are before b and d ?

(d) Choose an arbitrary pair of events from {A, B, C} above and determine if these two are independent or not.

## 1 Answer

For the first three, you can ignore all the letters except those referenced in the question. All permutations of the letters of interest are equally likely. Now you are down to few enough to count by hand if you don't have a better idea.

For the last, one choice of the pair makes it easy in light of my earlier paragraph.

• Thank you for your response! Unfortunately, I am not quite following what you are getting at in your first paragraph. – jewelltaylor9430 Apr 24 at 3:24
• For a, there are only two orders of $g$ and $a$. You can just erase all the other letters because you don't care about them. What is the chance $g$ is before $a$? – Ross Millikan Apr 24 at 3:31
• Okay I see, so the probability is 1/2 for a. By that same logic, is the probability of b 2/6 because there is 6 possible permutations and two of them have g after b and d? – jewelltaylor9430 Apr 24 at 3:35
• That is correct. Another way to think about it is to choose the last one, which is $g\ 1/3$ of the time. – Ross Millikan Apr 24 at 3:37
• Thanks for your help! You helped me me thinking about the question in the right way. – jewelltaylor9430 Apr 24 at 3:42