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Need help with problem solving

Hi. Can anyone help me solve this question below?

Five letters are selected at random from the 9 letters in the word ACTIVATED.

Find the probability that the selection does not contain more Ts than As.
Original post by hakewill12
Hi. Can anyone help me solve this question below?

Five letters are selected at random from the 9 letters in the word ACTIVATED.

Find the probability that the selection does not contain more Ts than As.

What have you tried?
Reply 2
I would divide the number of outcomes which contain more Ts than As by the total amount of outcomes and then subtract this from 1
My guess is you are having trouble counting how many selections fit the requirement.

Well, let's maybe start with what shape of selection must it be? So for instance, {A,A,T,T,_} is one of them.

You can go with Zarek's route of considering the compliment (i.e. what shape must the selection NOT be). You still need to go through the whole counting process though.

That said, if you aren't used to counting things like this, it is very easy to double count or not count cases. Please share your attempt, and we'll go from there.
Generally best to let the OP post some attempt, but most questions like this should be a few lines with the right counting method. As the order of the letters doesnt matter, you should be thinking about nCr, some way to handle the repetition of Ts and As.

So
#total = #2T2A + #2T1A + #2T0A + #1T2A + #1T1A + #1T0A + #0T2A + #0T1A + #0T0A
and each of the terms on the right can be calculated by thinking about how many combinations of the remaining letters can be chosen from the other 5 (not Ts or As). The required probability is then straightforward.

There are a few different ways you could do the counting, but having something clear is probably the main thing. But let the OP try first.
(edited 1 year ago)
Original post by mqb2766
Generally best to let the OP post some attempt, but most questions like this should be a few lines with the right counting method. As the order of the letters doesnt matter, you should be thinking about nCr, some way to handle the repetition of Ts and As.

So
#total = #2T2A + #2T1A + #2T0A + #1T2A + #1T1A + #1T0A + #0T2A + #0T1A + #0T0A
and each of the terms on the right can be calculated by thinking about how many combinations of the remaining letters can be chosen from the other 5 (not Ts or As). The required probability is then straightforward.

There are a few different ways you could do the counting, but having something clear is probably the main thing. But let the OP try first.

You are most probably right. I'm only a gcse maths student having a go. Really don't have much idea. I got rid of my workings. Can someone who knows how to do it please DM me the answer because I'm curious?
(edited 1 year ago)
Original post by hakewill12
Hi. Can anyone help me solve this question below?

Five letters are selected at random from the 9 letters in the word ACTIVATED.

Find the probability that the selection does not contain more Ts than As.


So if I do it right, you draw five letters in a row (out of 9) and after every single move the letters are not put back? If so, ask yourself these questions:

How many of these letters are A's and T's?
how many of them might be drawn in a draw?
what combinations of A's and T's are possible?
what combination is possible to have more A's than T's?
what is the total number of possible combinations?
(edited 1 year ago)
Couple of things to add:

This is a problem where you may want to "relabel" the A, T tiles as A1, A2 and T1, T2 to help you count correctly (e.g. a solution step counting the number of ways of getting "one A" has to consider that it could be A1 or A2).

As someone suggested quite early, to minimize the amount of work you're going to want to find 1 - p(more T's than A's).

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