Question about conditional probability

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Brushes

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Jul 19, 2021
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Hello.
I have a question about solving a specific conditional probability quesiton.

A person has 10 red hats and 10 pink hats.
He arranges the hats in 2 boxes.
In box 1 he puts 3 red hats and 7 pink hats.
In box 2 he puts 7 red hats and 3 pink hats.

The person tries to choose a hat, so he chooses one of the two boxes, and takes out 8 hats.

The question is, what is the probability that the seventh (7) hat is pink?

To solve this question I was thinking about using the probability tree, and check at which cases it's possible that the seventh hat is pink.
The problem that I encounter is that I deal with TONS of branches and cases to check.
Unfortunately our lecturer is unavailable for the next few days, so we can't ask him about the solution. Only on Sunday (two days from now) we'll be able to get an answer.
Is there any shorter way to check our case, instead of using a huge tree?

Maybe even just calculate the probability of picking 1 pink hat out of box no1, and the probability of picking 1 pink hat out of box no2, and sum them?


Thanks in advance.
 
Brushes said:
Hello.
I have a question about solving a specific conditional probability quesiton.

A person has 10 red hats and 10 pink hats.
He arranges the hats in 2 boxes.
In box 1 he puts 3 red hats and 7 pink hats.
In box 2 he puts 7 red hats and 3 pink hats.

The person tries to choose a hat, so he chooses one of the two boxes, and takes out 8 hats.

The question is, what is the probability that the seventh (7) hat is pink?

To solve this question I was thinking about using the probability tree, and check at which cases it's possible that the seventh hat is pink.
The problem that I encounter is that I deal with TONS of branches and cases to check.
Unfortunately our lecturer is unavailable for the next few days, so we can't ask him about the solution. Only on Sunday (two days from now) we'll be able to get an answer.
Is there any shorter way to check our case, instead of using a huge tree?

Maybe even just calculate the probability of picking 1 pink hat out of box no1, and the probability of picking 1 pink hat out of box no2, and sum them?


Thanks in advance.
Click to expand...
Yes, that last suggestion is right, if I understand it correctly.

Rather than take out 8 hats one at a time, imagine just laying the hats out in a row. The probability that the 7th hat in the row is pink is just the same as the probability that the first hat is pink; you might as well pick that one first.
 
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