Dear Stack Mathematicians,
I'm new to the site and I joined to ask a few math questions. I did some research regarding the topic, but I still have doubts.
Let's say I have a $4$-letters alphabet: {a, b, c ,d}, and I want to know the probability of getting a specific 5 digit string, e.g. 'aabcc'. Now, the possible combinations are $4^5$ , i.e. $1,024$. That means, if I pick up $5$ random letters (with replacement), I have 1/1024 chances of getting 'aabcc' (the order does matter). This means $\frac{1}{1024}$ $\approx0.001$ chance.
Now, I'm going to change things, let's say 'a' and 'b' have each $0.4$ chance to be chosen, while 'c' and 'd' have each $0.1$ chance to be chosen. The chance now to get 'aabcc' in that specific order will be: $0.4\cdot 0.4\cdot 0.4\cdot 0.1\cdot 0.1 = 0.00064$. Thus, if I generate $1500$ random $5$-letters words, I would expect (on average) one of them to be 'aabcc'. Is that correct?
I am looking forward to your feedback! :)