Calculating Probability of First Toss Being Tail in 3 Coin Tosses

[tictac123]

Qst A).

A coin is tossed 3 times and you note that the coin lands heads exactly twice. What is the probability that the first toss was the tail?


I know how to do it logically but not with all the notation.

Any help appreciated.

Thanks

 

[HallsofIvy]

That trick didn't work when you posted this in "homework"- it won't work here!

 

[mathman]

It looks like the answer should be 1/3, since there is nothing to favor anyone of the three tosses over any other.

 

[HallsofIvy]

If the problem had said "what is the probability the first coin was a tail" would you also answer "1/3"? Since "heads" or "tails" is all there can be, what does the remaining "1/3" probability represent?

tictac123, since you say you were able to do this "logically", what answer did you get?

 

[mathman]

If the problem had said "what is the probability the first coin was a tail" would you also answer "1/3"? Since "heads" or "tails" is all there can be, what does the remaining "1/3" probability represent?

tictac123, since you say you were able to do this "logically", what answer did you get?

Since the condition is that 2 tosses came up heads, the probability that the first toss was heads is 2/3.

 

[HallsofIvy]

There are 8 possible outcomes for three coins, and you can list them:
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Of those, exactly 4 have two "H"s, HHT, HTH, and THH. Of those 3, 2 have "H" first.

I would still like to know what tictac123 means by "I know how to do it logically but not with all the notation."

 

[NoMoreExams]

A solution exists?

 

[regor60]

There are 8 possible outcomes for three coins, and you can list them:
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Of those, exactly 4 have two "H"s, HHT, HTH, and THH. Of those 3, 2 have "H" first.

I would still like to know what tictac123 means by "I know how to do it logically but not with all the notation."

exactly "3" have two heads

 

[NoMoreExams]

exactly "3" have two heads

No 4 have 2 heads, HHH, HTH, HHT, THH. Out of them 3 have exactly 2 heads

 

[Niles]

It's 1/3. You can even use Bayes' sentence, but it might be overkill in this case since HallsofIvy's list says it all (but it's a good exercise to double-check it using Bayes' sentence).

 

[regor60]

Of those, exactly 4 have two "H"s, HHT, HTH, and THH. Of those 3, 2 have "H" first

No 4 have 2 heads, HHH, HTH, HHT, THH. Out of them 3 have exactly 2 heads

Who are you quibbling with, him or me ? He cites three as evidence of four. Obviously the issue is whether the intention was exactly two heads or two or more heads, but I'm just referring to his statement

 

[NoMoreExams]

Who are you quibbling with, him or me ? He cites three as evidence of four. Obviously the issue is whether the intention was exactly two heads or two or more heads, but I'm just referring to his statement

Probably you since to me "exactly 3 have 2 heads" doesn't mean exactly 2 heads since you put "exactly" before the 3.