# What is the expected number of flips to observe the first 5 flips pattern again?

• mmehdi
In summary, the problem at hand is to find the expected number of flips of a coin, with probability p of landing heads, until the first 5 flips are observed again. This includes the possibility of using some of the first flips. There are multiple ways to approach this problem, but finding a solution that works for all combinations of five flips is proving to be challenging. One potential solution involves setting up a linear difference equation and using a computer algebra package to solve it. However, it is also possible to turn it into a Bernoulli trial or use an indicator function variable. Overall, there is an intuitive feeling that there should be a straightforward closed form solution for this problem.
mmehdi
A coin having probability p of landing heads is flipped 5 times and then flipped utill the pattern of the first 5 flips is observed again for the first time, including the possiblity of using some of the first flips. If we want to find the expected number of flips of coins.(The lead is that: The expected number of additional flips after the five flips shall give us the answer)

...I have kind of tried to do it a couple of ways, but a solution which holds for all combination of five flips seems is proving to be too confounding.

Eek, where did this problem come from? The only method I see for solving it is setting up a giant linear difference equation for some auxiliary probabilities, thus allowing me to use a computer algebra package to solve the system and manipulate it to get the answer you want.

After the first five flips, every single flip shall lead to a new sequence of five flips. So the probability that the sequence shall terminate is whether every flip is such that it regenerates the earlier sequence. So I wonder if it could be turned into a burnoulli trial, or the expectation of an indicator function variable.

I have an intuitive feeling that this should have a rather straight forward close form solution.

Expected Number of flips

In order for the sequence to terminate on the sixth flip. The first five will have to be all heads or all tails. Meaning all the flips should be equal to the first flip. For the sequence to terminate in 7 flips the sequence should be (12121). Third flip equal to first flip, fourth equal to 2nd, fifth equal to 3rd, sixth equal to 4th and seventh equal to 5th.

For the sequence to terminate on the 8th flip the sequence will have to be (12312), 9 th flip (12341), and tenth flip (12345).

The problem came in an old exam, three years ago. One of the way to do it, is E[X]=P[X>1]+P[X>2]+P[X>3]...

Where P[X>1]= P[1-P[X=1]. Where P[X=1] which means termination at one additional flip is p^6(1-p)^6. I am just having trouble in extending it to infinite series

## 1. What is the expected number of coin flips to get heads?

The expected number of coin flips to get heads is 2 flips. This is because there are two possible outcomes (heads and tails) and each flip has a 50% chance of landing on heads.

## 2. How is the expected number of coin flips calculated?

The expected number of coin flips is calculated by taking the inverse of the probability of the desired outcome. In the case of flipping heads, this would be 1/0.5 = 2 flips.

## 3. Is the expected number of coin flips the same for each outcome?

No, the expected number of coin flips can vary for each outcome. For example, the expected number of flips to get tails is also 2 flips, but the expected number of flips to get both heads and tails in a row is 4 flips (2 flips for heads + 2 flips for tails).

## 4. Can the expected number of coin flips be used to predict the outcome of a single flip?

No, the expected number of coin flips is a statistical measure that gives the average number of flips needed to get a certain outcome over a large number of trials. It cannot be used to predict the outcome of a single flip.

## 5. How does the expected number of coin flips change with multiple coins?

The expected number of coin flips increases with multiple coins. For example, the expected number of flips to get at least one head in 2 coins is 3 flips, while in 3 coins it is 4 flips. This is because the probability of getting at least one heads decreases as the number of coins increases.

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