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Soledad

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Jan 16, 2022, 10:08 pm

Quote: Originally posted by cottoneyedjoe on Jan 16, 2022
The answer is 1/9.

Think of it like this. Pretend your friend is in another room of the house and he's playing with the two coins. He follows this procedure: shuffle the coins in his hand, choose one randomly, flip it three times. If he gets three heads he shouts out, "Hey I got three heads!" and repeats the procedure (shuffling the coins, randomly choosing one, flipping it three times.) If he gets any sequence other than three heads, he stays quiet and repeats the procedure. Pretend he does this for a long time so you have a big enough sample of shouts.

A majority of the times you hear him shouting, it's going to be that he flipped the coin with heads on both sides. A minority of the times you hear him shouting it will be that he flipped the fair coin. The proportion of the time that it's the fair coin is

(1/8) / (1 + 1/8) = 1/9

This is the from the conditional probability formula. The condition for hearing him shout is getting three heads in a row. The chance of that happening with the fair coin is 1/8, and the chance of that happening with the double-headed coin is 1.

Anyway, if you liked that problem, I can find some old 8th grade level Russian math olympiads problems to post later. And if you didn't like it, now you know to avoid my threads, lol.

Ok gimme a sec. I'm gonna reply to grwurston, and then maybe you can reply to me. You have me a little confused with your 1/8

Sometimes you do the right thing just because it’s right.

Life's a game of mathematics, if you can't figure that out I don't know what to tell you.
Soledad

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Jan 16, 2022, 10:12 pm

Quote: Originally posted by grwurston on Jan 16, 2022
So then what is it?

Ok, half of the time it is expected you pick the fair coin and half of the time the trick coin, hence 50/50. When you pick the fair coin, half of the times you get heads. When you pick the trick coin all of the time you get heads. Out of the 4 chances of getting heads, 2 sides to each coin, 3 out of 4 you will get heads. So out of 100 chances, 25 chances it was the fair coin, and 50 chances it was the trick coin. So 1/3 I would say.

Sometimes you do the right thing just because it’s right.

Life's a game of mathematics, if you can't figure that out I don't know what to tell you.
cottoneyedjoe

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Jan 16, 2022, 11:24 pm

Quote: Originally posted by Soledad on Jan 16, 2022
Ok, half of the time it is expected you pick the fair coin and half of the time the trick coin, hence 50/50. When you pick the fair coin, half of the times you get heads. When you pick the trick coin all of the time you get heads. Out of the 4 chances of getting heads, 2 sides to each coin, 3 out of 4 you will get heads. So out of 100 chances, 25 chances it was the fair coin, and 50 chances it was the trick coin. So 1/3 I would say.

This is almost correct except you have to use the probability of getting three heads on three flips with a fair coin, not the overall probability of getting heads on any flip. So you get 0.5 x 0.5 x 0.5 = 0.125 = 1/8 as the probability of three heads in three flips of the fair coin. (The part about flipping it three times isn't extraneous information in the statement of the problem.)
CARBOB

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Jan 19, 2022, 9:04 am

Very good, always give examples! Helps to clarify!!!
Speler

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Feb 12, 2022, 6:05 pm

Quote: Originally posted by cottoneyedjoe on Jan 16, 2022
The answer is 1/9.

Think of it like this. Pretend your friend is in another room of the house and he's playing with the two coins. He follows this procedure: shuffle the coins in his hand, choose one randomly, flip it three times. If he gets three heads he shouts out, "Hey I got three heads!" and repeats the procedure (shuffling the coins, randomly choosing one, flipping it three times.) If he gets any sequence other than three heads, he stays quiet and repeats the procedure. Pretend he does this for a long time so you have a big enough sample of shouts.

A majority of the times you hear him shouting, it's going to be that he flipped the coin with heads on both sides. A minority of the times you hear him shouting it will be that he flipped the fair coin. The proportion of the time that it's the fair coin is

(1/8) / (1 + 1/8) = 1/9

This is the from the conditional probability formula. The condition for hearing him shout is getting three heads in a row. The chance of that happening with the fair coin is 1/8, and the chance of that happening with the double-headed coin is 1.

Anyway, if you liked that problem, I can find some old 8th grade level Russian math olympiads problems to post later. And if you didn't like it, now you know to avoid my threads, lol.

I heard that one in class, but don't think that it is written in my course. Same for the calculation of PI and alike.
Speler

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Feb 12, 2022, 6:09 pm

Quote: Originally posted by cottoneyedjoe on Jan 13, 2022
Middle school math competition question:

You have two coins: one fair, and one that is heads on both sides. You randomly choose one of the coins without looking at it and flip it three times in a row. It comes out heads each time. (Assume you don't look at the coin except when it lands and have no other information). What is the probability it's the fair coin?

The probability of picking the fair coin is 0.5.
Speler

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Feb 12, 2022, 6:14 pm

(0.5*1/8)+0.5
=0.5625
(0.5*1/8)/0.5625
=0.1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
0.5625/(0.5*1/8)
=9

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