# What Is the Probability of a Third Head After Two Heads with a Biased Coin?

• davidmoore63@y
In summary, a biased coin is constructed with a probability of p for heads and 1-p for tails. After flipping the coin twice and getting two heads, the probability of a third head is 3/4. This also means the fair value of a lottery ticket that pays one dollar for a third head is 3/4, and cannot be sold for an irrational number's worth of currency.
davidmoore63@y
A random number p such that 0<p<1 is selected at random from a uniform distribution U[0,1]. A biased coin is then constructed such that the probability of heads on a single flip is p (thus 1-p for a tails).

This coin is flipped twice and the result is HH. If the coin is flipped a third time, what is the probability of a third head? More precisely, what is the fair value of a lottery ticket that pays one dollar if the third flip is a head, and zero otherwise? What would you pay for it/ sell it for?

Is this a trick question?

"A biased coin is then constructed such that the probability of heads on a single flip is p..."

not at all! I think it's well defined isn't it?

Then the two H results don't give you any additional information. You know the probability is p, and you know the expected value of a $1 bet is p x$1.

The question is asking for the probability of a third head PRIOR to finding out what p is.

the bet is worth \$0 as the probability of drawing a rational number from the uniform distribution is zero and you can't pay someone and irrational number's worth of currency ;)

Ok for you we round p to the nearest 1/100. You still have to do the question now!

We want to know the probability of HHH given HH. It's Prob(HHH) / Prob(HH). The probability of HHH is the expectation value of p^3 where p is uniformly distributed on the interval [0, 1]. So it's int_0^1 p^3 dp = 1/4. Similarly Prob(HH) = int_0^1 p^2 dp = 1/3. So the answer is 3/4.

Looks right to me

## 1. What is the probability of getting heads or tails when flipping a coin?

The probability of getting either heads or tails when flipping a coin is 50%. This is because there are only two possible outcomes, and each outcome has an equal chance of occurring.

## 2. Is it possible to predict the outcome of a coin flip?

No, it is not possible to predict the outcome of a coin flip with certainty. The outcome is completely random and cannot be influenced by any external factors.

## 3. Can the probability of getting heads or tails change over time?

No, the probability of getting heads or tails remains constant at 50% for each individual coin flip. However, if a large number of flips are performed, the overall percentage of heads and tails may not be exactly 50% due to chance.

## 4. Is there a way to increase the chances of getting a certain outcome when flipping a coin?

No, there is no way to increase the chances of getting a certain outcome when flipping a coin. The outcome is always 50% for each possibility and cannot be manipulated.

## 5. How many times should a coin be flipped to get an accurate representation of the probability?

The more times a coin is flipped, the more accurate the representation of the probability will be. However, it is generally accepted that at least 100 flips should be performed to get a reasonable estimate of the probability.

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