Probability question on fair coin

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In summary, the probability of the pattern T,H occurring before the pattern H,H in a sequence of coin flips is 3/4. This can be calculated by first determining the probability of getting HH before TH, which is p^2, and then subtracting that from 1 to get 1-p^2. Since the coin is fair, p=1/2, so 1- (1/2)^2 = 3/4.
  • #1
ichabodgrant
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Homework Statement


A fair coin is continually flipped. What is the probability that the pattern T,H occurs before the pattern H,H, where T and H respectively denote Tail and Head of a coin?

Homework Equations


Prob. = (n r) (pr)(1-p)n-r

The Attempt at a Solution


I am thinking whether the question asks about:

i. prob. of TH vs HT vs TT vs HH
ii. THHHHHHH...or TTHHHHHH... or HTHHHHHH... or HTTHHHHH... or HTTTTTHHH... or ...

If it is the 2nd one, how can we calculate the prob.? It looks like an infinitely long series... or should I let there be n trials? Then I use binomial distribution to find?
 
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  • #2
Is there any reason why one of the two results is more likely than the other?
 
  • #3
I don't quite understand your question...
The prob. of getting a head or a tail is 1/2? Is this what you are asking?
 
  • #4
ichabodgrant said:
I don't quite understand your question...
The prob. of getting a head or a tail is 1/2? Is this what you are asking?
I'm asking whether you can think of any reason why getting HT is more or less likely than getting TH.
 
  • #5
Same?
 
  • #6
ichabodgrant said:
Same?
Yes.
 
  • #7
So simply calculate the first case?
 
  • #8
ichabodgrant said:
So simply calculate the first case?
What will the two probabilities add up to?
 
  • #9
Which two? TH and HT?

0.5*0.5 + 0.5*0.5 = 0.5?
 
  • #10
ichabodgrant said:
Which two? TH and HT?

0.5*0.5 + 0.5*0.5 = 0.5?
No.
The two events are,
  • that TH occurs before HT in an arbitrarily long sequence,
  • that HT occurs before TH in an arbitrarily long sequence
What must those two probabilities add up to?
 
  • #11
1?
 
  • #12
but it seems 0.5*0.5*0.5*... , gets you 0...
 
  • #13
ichabodgrant said:
1?
Yes.
ichabodgrant said:
but it seems 0.5*0.5*0.5*... , gets you 0...
What exactly is the event for which that calculation applies?
 
  • #14
Oh Sorry...

I have a typo in the question...
The correct question is "T,H before HH"
 
  • #15
Should I indeed use conditional probability?
In case I have a T at the 1st trial, then it already achieves the event that TH appears before HH?
 
  • #16
ichabodgrant said:
The correct question is "T,H before HH"
Ok, that makes it a lot more interesting.
Assign unknowns to those two probabilities. Consider the first two tosses. There are four situations at that point, equally likely. Consider the probabilities of the two outcomes of interest when continuing from each of those four positions.
See what equations you can extract.
 
  • #17
Actually TT, HT,TH all means TH must occur before HH, right?
 
  • #18
so if there is no HH, then it is alright?
Let the probability of getting a H be p.

Then the required prob. is 1 - p2?
 
  • #19
ichabodgrant said:
so if there is no HH, then it is alright?
What do you mean "it's alright"?
There are four equally likely states after two tosses:
HH
HT
TH
TT
In two of those, the outcome is already determined, yes?
Look at the remaining two. Can you predict what the eventual outcome will be for those?
 
  • #20
If it is HT, then no matter the next one is T or H, TH already occurs before HH, right?
If it is TT, then also no matter the next one is T or H, TH already occurs before HH, right?
If it is TH, then obviously no matter the next one is T or H, TH already occurs before HH, right?
Just the case if you obtain all H and no T at the first 2 tosses will give you HH before TH..

I am thinking in this way
 
  • #21
ichabodgrant said:
If it is HT, then no matter the next one is T or H, TH already occurs before HH, right?
If it is TT, then also no matter the next one is T or H, TH already occurs before HH, right?
If it is TH, then obviously no matter the next one is T or H, TH already occurs before HH, right?
Just the case if you obtain all H and no T at the first 2 tosses will give you HH before TH..

I am thinking in this way
Yes.
 
  • #22
so the probability of getting HH is p2?
Then the required probability is 1- p2?
 
  • #23
ichabodgrant said:
so the probability of getting HH is p2?
Then the required probability is 1- p2?
Yes, but can't you assume it's a fair coin?
 
  • #24
then 1 - 0.52 = 3/4
 
  • #25
thank you for your kindness
 

What is a fair coin?

A fair coin is a type of coin that has an equal chance of landing on either heads or tails when flipped. This means that the probability of getting heads is the same as getting tails, making it a balanced and unbiased coin.

What is the probability of getting heads or tails on a fair coin?

The probability of getting heads or tails on a fair coin is 50%, or 1/2. This means that if you were to flip a fair coin multiple times, you would expect to get heads roughly half of the time and tails the other half.

What is the probability of getting a certain number of heads or tails in a series of coin flips?

The probability of getting a specific number of heads or tails in a series of coin flips can be calculated using the binomial distribution formula. This formula takes into account the number of trials, the probability of success (getting heads or tails), and the number of desired outcomes.

What is the difference between theoretical probability and experimental probability in the context of a fair coin?

Theoretical probability is the probability of an event occurring based on mathematical calculations and assumptions, while experimental probability is the probability of an event occurring based on actual trials or experiments. In the context of a fair coin, the theoretical probability of getting heads or tails is 50%, while the experimental probability may vary slightly depending on the actual outcomes of the coin flips.

How does the law of large numbers apply to the probability of a fair coin?

The law of large numbers states that as the number of trials or experiments increases, the experimental probability will approach the theoretical probability. This means that the more times you flip a fair coin, the closer the percentage of heads and tails will get to 50%, demonstrating the fairness of the coin.

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