How to compare the probability of the following situation?

A letter is drawn 1000 times from A, A, A, R, B, I.

1) You win a dollar if the number of A’s among the draws is 10 times or more above the expected number.

2) You win a dollar if the number of B’s among the draws is 10 times or more above the expected number.

Intuitively, I feel the answer is A because we have a better chance pick A.
But I am not quite sure about the standard way to think about it. Could someone suggest a direction?

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Do you mean “10 times or more” as in x+10x+10 or as in 10x10x?
– suomynonA
Oct 21 at 2:21

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1 Answer
1

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What is the probability of drawing an A? 12.\frac1{2}.

What is the probability of drawing a B? 16.\frac1{6}.

What is the expected number of A’s in 10001000 draws? 500.500.

What is the expected number of B’s in 10001000 draws? 16623.166 \frac2{3}.

What is the 1010 times the expected number of A’s in 10001000 draws? 5000.5000.

What is the 1010 times the expected number of B’s in 10001000 draws? 166623.1666 \frac2{3}.

Now you can answer these:

What is the probability of getting 50005000 A’s or more in 10001000 draws?

What is the probability of getting 16671667 B’s or more in 10001000 draws?

Hint: Don’t take either of these bets!