# Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs.

What is the expected number of coin tosses?

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Yet another copy and paste from Brilliant.org: brilliant.org/i/5rCgJ3 – Erick Wong Apr 17 '13 at 5:49
@ErickWong: Is this a recent problem on brilliant.org? – robjohn Apr 17 '13 at 8:48

Let $e$ be the expected number of tosses. It is clear that $e$ is finite.

Start tossing. If we get a tail immediately (probability $\frac{1}{2}$) then the expected number is $e+1$. If we get a head then a tail (probability $\frac{1}{4}$), then the expected number is $e+2$. Continue $\dots$. If we get $4$ heads then a tail, the expected number is $e+5$. Finally, if our first $5$ tosses are heads, then the expected number is $5$. Thus $$e=\frac{1}{2}(e+1)+\frac{1}{4}(e+2)+\frac{1}{8}(e+3)+\frac{1}{16}(e+4)+\frac{1}{32}(e+5)+\frac{1}{32}(5).$$ Solve this linear equation for $e$. We get $e=62$.

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It is clear that e is finite, but how can you show it properly though ? Thanks. – Dark Jul 3 '15 at 17:39
If one wants, let $X$ be the number of tosses. Then $\Pr(X=n)\le (1/2)^{n-5}$. So $E(X)\le \sum n (1/2)^{n-5}$, a convergent series. – André Nicolas Jul 3 '15 at 17:58
The same method obviously generalizes to give $e_n$, the expected number of tosses to get $n$ consecutive heads ($n \ge 1$): $$e_n=\frac{1}{2}(e_n+1)+\frac{1}{4}(e_n+2)+\frac{1}{8}(e_n+3)+\frac{1}{16}(e_n+‌​4)+\cdots +\frac{1}{2^n}(e_n+n)+\frac{1}{2^n}(n),$$ the solution of which is easily found to be $$e_n = 2(2^n - 1).$$ – r.e.s. Jul 19 '15 at 23:57
Yes, good point. The most common assigned problem is expected number of tosses until two heads in a row, answer $6$. We can use the same idea for $n$ heads in a row, probability of head $p$. – André Nicolas Jul 20 '15 at 0:12
Can you help me to find out what is the expected number of tossing if the probability of getting head, $p\neq0.5$? Is it $$E_n=np^n+\sum_{k=1}^n p^k(E_n+k)?$$ – Anastasiya-Romanova 秀 May 16 at 2:37

Here is a generating function approach.

Consider the following toss strings, probabilities, and terms

$$\color{#00A000}{ \begin{array}{llc} T&\frac12&\qquad\frac12x\\ HT&\frac14&\qquad\frac14x^2\\ HHT&\frac18&\qquad\frac18x^3\\ HHHT&\frac1{16}&\qquad\frac1{16}x^4\\ HHHHT&\frac1{32}&\qquad\frac1{32}x^5\\ \color{#C00000}{HHHHH}&\color{#C00000}{\frac1{32}}&\color{#C00000}{\qquad\frac1{32}x^5} \end{array} }$$ Each term has the probability as its coefficient and the length of the string as its exponent.

Possible outcomes are any combination of the green strings followed by the red string. We get the generating function of the probability of ending after $n$ tosses to be \begin{align} f(x)&=\sum_{k=0}^\infty\left(\frac12x+\frac14x^2+\frac18x^3+\frac1{16}x^4+\frac1{32}x^5\right)^k\frac1{32}x^5\\ &=\frac{\frac1{32}x^5}{1-\left(\frac12x+\frac14x^2+\frac18x^3+\frac1{16}x^4+\frac1{32}x^5\right)}\\ &=\frac{\frac1{32}x^5}{1-\frac{\frac12x-\frac1{64}x^6}{1-\frac12x}}\\ &=\frac{\frac1{32}x^5-\frac1{64}x^6}{1-x+\frac1{64}x^6} \end{align} The average duration is then \begin{align} f'(1) &=\left.\frac{\left(\frac5{32}x^4-\frac6{64}x^5\right)\left(1-x+\frac1{64}x^6\right)-\left(\frac1{32}x^5-\frac1{64}x^6\right)\left(-1+\frac6{64}x^5\right)}{\left(1-x+\frac1{64}x^6\right)^2}\right|_{\large x=1}\\ &=\frac{\frac4{64}\frac1{64}+\frac1{64}\frac{58}{64}}{\left(\frac1{64}\right)^2}\\[12pt] &=62 \end{align}

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Could you elaborate briefly on why the derivative gives the expected number of flips? – Austin Mohr Aug 20 '13 at 2:37
@AustinMohr: If $f(x)$ is the generating function of the probability $p_n$ of the ending after $n$ tosses $$f(x)=\sum_{n=0}^\infty p_nx^n$$ then, because the probability of lasting an infinite number of tosses is $0$, we have \begin{align} f(1) &=\sum_{n=0}^\infty p_n\\ &=1 \end{align} Furthermore, \begin{align} f'(1) &=\sum_{n=0}^\infty n\,p_n\\ &=\mathrm{E}(n) \end{align} – robjohn Aug 20 '13 at 5:13
Do you know how to find the distribution (or expectation and variance) for the number of tosses until either 5 consecutive heads or 5 consecutive tails? (Or 5 consecutive equal results from rolling dice.) Is there a question on math.se about this? – ShreevatsaR Dec 15 '15 at 14:17
I found an answer using martingales here: quora.com/… but I'm curious if there is a generating functions way (also about the distribution, say variance or number of trials until 90% probability of seeing what we want). – ShreevatsaR Dec 15 '15 at 14:31
@ShreevatsaR: The generating function for the probability of ending on $n$ tosses for that problem is similar: $$\frac{\frac1{16}x^5}{1-\left(\frac12x+\frac14x^2 +\frac18x^3+\frac1{16}x^4\right)}$$ From that, we can compute the expectation and variance. Looking at the coefficients of the series for the generating function we can also find out that in $66$ tosses, we will have a $90.0761\%$ chance of seeing $5$ heads or $5$ tails in a row. – robjohn Dec 15 '15 at 17:32

This problem is solvable with the next step conditioning method. Let $\mu_k$ denote the mean number of tosses until 5 consecutive heads occurs, given that $k$ consecutive heads just occured. Obviously $\mu_5=0$. Conditioning on the outcome of the next coin throw: $$\mu_k = 1 + \frac{1}{2} \mu_{k+1} + \frac{1}{2} \mu_0$$ Solving the resulting linear system:

In[28]:= Solve[Table[mu[k] == 1 + 1/2 mu[k + 1] + mu[0]/2, {k, 0, 4}],
Table[mu[k], {k, 0, 4}]] /. mu[5] -> 0

Out[28]= {{mu[0] -> 62, mu[1] -> 60, mu[2] -> 56, mu[3] -> 48,
mu[4] -> 32}}


Hence the expected number of coin flips $\mu_0$ equals 62.

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What tool did you use for solving? – pushpen.paul May 11 '15 at 17:19
@pushpen.paul I used Mathematica – Sasha May 11 '15 at 17:20

Lets calculate it for $n$ consecutive tosses the expected number of tosses needed.

Lets denote $E_n$ for $n$ consecutive heads. Now if we get one more head after $E_{n-1}$, then we have $n$ consecutive heads or if it is a tail then again we have to repeat the procedure.

So for the two scenarios:

1. $E_{n-1}+1$
2. $E_{n}{+1}$ ($1$ for a tail)

So, $E_n=\frac12(E_{n-1} +1)+\frac12(E_{n-1}+ E_n+ 1)$, so $E_n= 2E_{n-1}+2$.

We have the general recurrence relation. Define $f(n)=E_n+2$ with $f(0)=2$. So,

\begin{align} f(n)&=2f(n-1) \\ \implies f(n)&=2^{n+1} \end{align}

Therefore, $E_n = 2^{n+1}-2 = 2(2^n-1)$

For $n=5$, it will give us $2(2^5-1)=62$.

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The question can be generalized to what is the expected number of tosses before we get x heads.Let's call this E(x). We can easily derive a recursive formula for E(x). Now, there are a total of two possibilities, first is that we fail to get the xth consecutive heads in xth attempt and second, we succeed. Probability of success is 1/(2^x) and probability of failure is 1-(1/(2^x)).

Now, if we were to fail to get xth consecutive heads in xth toss (i.e. case 1), the we will have to use a total of (E(x)+1) moves, because one move has been wasted.

On the other hand if we were to succeed in getting xth consecutive head in xth toss (i.e. case 2), the total moves is E(x-1)+1 , because we now take one move more than that was required to get x-1 consecutive heads.

So,

E(x) = P(failure) * (E(x)+1) + P(success) * (E(x-1)+1)
E(x) = [1-(1/(2^x))] * (E(x)+1) + [1/(2^x)] * (E(x-1)+1)

Also E(0) = 0 , because expected number of tosses to get 0 heads is zero, duh

now,

E(1) = (1-0.5) * (E(1)+1) + (0.5) * (E(0)+1) => E(1) = 2

E(2) = (1-0.125) * (E(1)+1) + (0.125) * (E(1)+1) => E(2) = 6

Similarly,

E(3) = 14

E(4) = 30

E(5) = 62

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Expectation for getting n consecutive heads is : 2*(2^n-1). Thus for 5 heads it is = 62.

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How did you get the formula? – pushpen.paul May 11 '15 at 18:25

I would simplify the problem as follows:

Let $e$ = Expected number of flips until $5$ consecutive $H$, i.e., $E[5H]$
Let $f$ = Expected number of flips until $5$ consecutive $H$ when we have seen one $H$, i.e., $E[5H|H]$
Let $g$ = Expected number of flips until $5$ consecutive $H$ when we have seen two $H$, i.e., $E[5H|2H]$
Let $h$ = Expected number of flips until $5$ consecutive $H$ when we have seen three $H$, i.e., $E[5H|3H]$
Let $i$ = Expected number of flips until $5$ consecutive $H$ when we have seen four $H$, i.e., $E[5H|4H]$

Now Start flipping coin, there is $\frac{1}{2}$ probability of getting $H$ or $T$. So if we get $H$ then expected number of flips until 5 consecutive $H$ is $(f+1)$. Alternatively if $T$, we wasted 1 flip and expected number is still $(e+1)$ $$e=\frac12(e+1)+\frac12(f+1)\;$$

We now need $f$ to solve above to get $e$. Now we start with 1 $H$ and seeking 4 more $H$ to get total 5 $H$. Again, there is $\frac{1}{2}$ probability of getting $H$ or $T$. So if we get $H$ (total $2H$ so far) then expected number of flips until 5 consecutive $H$ is $(g+1)$. Alternatively if $T$, we wasted this flip and expected number is back to $(e+1)$

$$f=\frac12(g+1)+\frac12(e+1)\;$$

Continuing this way... $$g=\frac12(h+1)+\frac12(e+1)\;$$ $$h=\frac12(i+1)+\frac12(e+1)\;$$

Finally, Now we have 4 $H$ and seeking last $H$ to get total 5 $H$. Still, there is $\frac{1}{2}$ probability of getting $H$ or $T$. So if we get $H$ (total $5H$) then we need just $1$ flip. Alternatively if $T$ is observed, we wasted this flip and expected number is back to $(e+1)$ $$i=\frac12(1)+\frac12(e+1)\;$$

Solving these equations, $e=62$, $f=60$, $g=56$, $h=48$, $i=32$
This solution offers some insight into conditional expectations of number of flips needed till 5 consecutive $H$ given 1, 2, 3 and 4 consecutive $H$.

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No one else seems to have suggested the following approach. Suppose we keep flipping a coin until we get five heads in a row. Define a "run" as either five consecutive heads or a tails flip plus the preceding streak of heads flips. (A run could be a single tails flip.) The number of coin flips is equal to the number of runs with at least four heads ($R_{4+}$), plus the number of runs with at least three heads ($R_{3+}$), and so on down to the number of runs with at least zero heads ($R_{0+}$). We can see this by expanding the terms:

$R_{0+}$ = # runs with 0 heads + # runs with 1 head + ... + # runs with 5 heads

$R_{1+}$ = # runs with 1 head + # runs with 2 heads + ... + # runs with 5 heads

...

$R_{4+}$ = # runs with 4 heads + # runs with 5 heads

# flips = # flips in runs with 0 heads + # flips in runs with 1 head + ... + # flips in runs with 5 heads

# flips in runs with 0 heads = # runs with 0 heads

# flips in runs with 1 head = 2 x # runs with 1 head

...

# flips in runs with 4 heads = 5 x # runs with 4 heads

# flips in runs with 5 heads = 5 x # runs with 5 heads

By linearity of expectation, the expected number of coin flips is $E(R_{0+}) + E(R_{1+}) + \ldots + E(R_{4+})$. $E(R_{4+})$ is $2 E(R_{5+}) = 2$, because one half of the time we flip at least four heads in a row, we go on to flip five heads in a row, i.e. the following coin flip is heads. In other words, in expectation, it takes two runs that start with four heads to achieve one run of five heads. Likewise, $E(R_{3+}) = 2 E(R_{4+}) = 4$, $E(R_{2+}) = 2 E(R_{3+}) = 8$, $E(R_{1+}) = 2 E(R_{2+}) = 16$, and $E(R_{0+}) = 2 E(R_{1+}) = 32$. The expected number of coin flips is $32 + 16 + 8 + 4 + 2 = 62$.

More generally, given a biased coin that comes up heads $p$ portion of the time, the expected number of flips to get $n$ heads in a row is $\frac{1}{p} + \frac{1}{p^2} + \ldots + \frac{1}{p^n} = \frac{1 - p^n}{p^n(1 - p)}$.

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