# The Probability of a Biased Coin: n Flips, m Heads

• MHB
• markosheehan
In summary, the probability of getting a head on flipping a biased coin is p. the coin is flipped n times producing a sequence containing m heads and (n-m) tails what is the probability of obtaining this sequence from n flips.
markosheehan
the probability of getting a head on flipping a biased coin is p. the coin is flipped n times producing a sequence containing m heads and (n-m) tails what is the probability of obtaining this sequence from n flips.
i can't understand the wording

I've moved this thread since our advanced forum is for calculus based stats.

A few things we need to observe:

The probability of getting heads is:

$$\displaystyle P(H)=p$$

Now, we know that it is certain that we will either get heads or tails, so we may state:

$$\displaystyle P(H)+P(T)=1\implies P(T)=1-P(H)=1-p$$

So, the probability of getting $m$ heads is:

$$\displaystyle P\left(H_m\right)=p^m$$

And the probability of getting $n-m$ tails is:

$$\displaystyle P\left(T_{n-m}\right)=(1-p)^{n-m}$$

Next we need to look at the number $N$ of ways to choose $m$ from $n$:

$$\displaystyle N={n \choose m}$$

Can you put all this together to find the requested probability?

when i put this all together i get (n ncr m)*p*(1-p)^n-m however at the back of the book it says the answer is p^m(1-p)^n-m

What I get is:

$$\displaystyle P(X)={n \choose m}p^m(1-p)^{n-m}$$

And this agrees with the binomial probability formula. :D

This is the probability of getting any sequence with $m$ heads, for any particular such sequence, then it would be:

$$\displaystyle P(X)=p^m(1-p)^{n-m}$$

markosheehan said:
when i put this all together i get (n ncr m)*p*(1-p)^n-m however at the back of the book it says the answer is p^m(1-p)^n-m
Was it possible that the problem asked for the probability of m heads in a row followed by n-m tails in a row? As MarkFl said, that probability if for any particular such sequence- "m heads in a row followed by n- m tails in a row" or "n- m tails in a row followed by m heads in a row" or "A head, then a tail, then a head, followed by m- 2 heads in a row, followed by n- m- 1 tails in a row", etc.

## 1. What is a biased coin?

A biased coin is a coin that has a higher probability of landing on one side compared to the other. This means that the outcomes of the coin toss are not equally likely.

## 2. How do you determine the probability of a biased coin?

The probability of a biased coin can be determined by dividing the number of desired outcomes (such as heads) by the total number of possible outcomes. For example, if a coin is biased to land on heads 70% of the time, the probability of getting heads is 0.7 or 70%.

## 3. How does the number of flips affect the probability of getting m heads?

The more flips you do, the closer the actual results will be to the expected probability. For example, if a coin is biased to land on heads 70% of the time, the more flips you do, the closer you will get to 70% of the flips resulting in heads.

## 4. Can the probability of a biased coin change over time?

Yes, the probability of a biased coin can change over time. This can happen due to external factors such as wear and tear on the coin, or if the coin is intentionally altered.

## 5. How does the probability of a biased coin compare to a fair coin?

The probability of a biased coin will always be different from a fair coin, where the probability of getting heads or tails is equal. A fair coin has a probability of 0.5 or 50% for both heads and tails, while a biased coin will have different probabilities for each outcome.

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