Calculating Probability of First Toss Being Tail in 3 Coin Tosses

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Discussion Overview

The discussion revolves around calculating the probability that the first toss of a coin is a tail, given that a coin is tossed three times and lands heads exactly twice. The conversation explores different approaches to the problem, including logical reasoning and probabilistic notation.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asks for help with the probability calculation, expressing familiarity with logical reasoning but difficulty with notation.
  • Another participant suggests that the probability should be 1/3, arguing that there is no reason to favor any of the three tosses over the others.
  • A different participant questions whether the answer would still be 1/3 if the question were framed differently, pointing out that the condition of having two heads affects the probabilities.
  • One participant lists the possible outcomes of three coin tosses and notes that among the outcomes with two heads, two have heads first, suggesting a different probability calculation.
  • There is a correction regarding the number of outcomes with two heads, with some participants asserting that there are four such outcomes while others argue about the interpretation of "exactly" two heads.
  • Another participant mentions the use of Bayes' theorem as a potential method for solving the problem, though they suggest it may be unnecessary given the listed outcomes.
  • Disagreement arises over the correct interpretation of the number of outcomes with two heads, with participants clarifying their positions on the counts and definitions used.

Areas of Agreement / Disagreement

Participants express differing views on the correct probability calculation and the interpretation of the outcomes. There is no consensus on the final answer, and the discussion remains unresolved with competing perspectives on the problem.

Contextual Notes

Participants highlight ambiguities in the phrasing of the problem, particularly regarding the distinction between "exactly" two heads versus "two or more" heads, which affects the probability calculations. There are also unresolved mathematical steps in the reasoning presented.

tictac123
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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
 
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That trick didn't work when you posted this in "homework"- it won't work here!
 


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


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?
 


HallsofIvy said:
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.
 


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."
 


A solution exists?
 


HallsofIvy said:
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
 


regor60 said:
exactly "3" have two heads

No 4 have 2 heads, HHH, HTH, HHT, THH. Out of them 3 have exactly 2 heads
 
  • #10


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).
 
  • #11


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


NoMoreExams said:
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
 
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  • #12


regor60 said:
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.
 

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