Question

Say we flip a fair coin n (for example n=100) times. How would one calculate the expected value of how many time a sequence occurs (for example HHT where H:heads, T:tails)?

Answer 1

Can't figure this problem out other than that it would be the same for any sequence that the expected value would be: \[n\times \frac{1}{2^3}\]

Answer 2

How did you get 98? How would that change if the sequence was for example HHH?

Answer 3

i have a few other ideas to try so i ask

Answer 4

I understand your solution but It's a part of a problem problem involving Penney's game which I know isn't a fair game thus I would expect different expectations for different sequences?

Answer 5

or is it only when comparing relative probabilities of different sequences one would find different probabilities/expectations for different sequences?

Answer 6

dan i was thinking maybe usin combinations or permutations
not too sure tho

Answer 7

thats hwat i mean

Answer 8

for example you look at the first 3 in the 100 flips, are you allowed then look at the next 3 starting from the 2nd one, and say that is also 1/8 , i dont think u can because it is included when you checked the first 3

Answer 9

my other method was

Answer 10

lets check 3 at a time differently

Answer 11

sigh maybe i shouldht have deleted that stuff, it doesnt seem wrong

Answer 12

oh nvm it is wrong!

Answer 13

i see what u mean
if they are all HH you can have a chance of 98 sequences
but there is no way to have more than a sequence of 33 hot HHT

Answer 14

do you know what hte right answer is?

Answer 15

Unfortunately there are no answers to the questions but I can ask my professor next week. I really appreciate you taking the time to discuss my question, thank you.

Answer 16

wait lol

Answer 17

I actually can get the right answer

Answer 18

we can work backwards

Answer 19

iam writing a program for it right now

Answer 20

oh, cool, what do you mean with working backwards?

Answer 21

like see what the probability really is

Answer 22

and then we can try some ideas and see which one is actually giving us the right answer

Answer 23

okay done!

Answer 24

ok

Answer 25

nice :)

Answer 26

i will run it for 100 times

Answer 27

100 times

Answer 28

and see what hte average is

Answer 29

then we will see for 1000 flips and avg that over 100 times

Answer 30

ok, interesting

Answer 31

okay i got

Answer 32

avgc()
11.83
>>> avgc()
11.87
>>> avgc()
11.88
>>> avgc()
11.78
>>> avgc()
12.15

Answer 33

avg
12.23

Answer 34

always hobering around 12

Answer 35

12 HHT for 100 flips

Answer 36

lets see if the ratio remains for 1000 flips

Answer 37

avgc()
124.833
>>> avgc()
124.711
>>> avgc()
124.499

Answer 38

the 1000 one is a lot more accurate

Answer 39

always hovering around 124.5

Answer 40

so for 100 we should be getting something like 12.45

Answer 41

avgc()
124.833
>>> avgc()
124.711
>>> avgc()
124.499
>>> avgc()
124.507
>>> avgc()
124.546
>>> avgc()
124.719
>>> avgc()
124.792

Answer 42

yep

Answer 43

ok, is it the same result for other sequences?

Answer 44

nope

Answer 45

what sequence you want to try ?

Answer 46

do you know how to use python

Answer 47

Somewhat, had an introductory course to programming in python.

Answer 48

for example HHH

Answer 49

i did avg of the 100 range 100 times and it is always 12.1 ish

Answer 50

here is my code

Answer 51

on the 9th line you can change it any sequence you want

Answer 52

as long as its length 3

Answer 53

i will try HHH

Answer 54

Thank you, I'll try it

Answer 55

oh its actually the same for HHH :O

Answer 56

samething for htt

Answer 57

12.1 to 12.15 ish

Answer 58

you can improve the accuracy if you tweak final part of the code

Answer 59

if you make the last part range 1000 and divided by 1000

Answer 60

but it might slightly lagg your computer

Answer 61

ok, i'll try it. Thanks for the program! What led me to believe that the expectation was different was the analysis in this wikipedia article: http://en.wikipedia.org/wiki/Penney's_game#Analysis_of_the_three-bit_game

Answer 62

wait a second let me check my code!!

Answer 63

it looks right

Answer 64

the brute force method cant be that off -_- unless i made some mistake in code let me check again

Answer 65

from random import*
def check(a):
x=['h','t']
slist=[]
for i in range(a):
slist.append(x[randint(0,1)])

count = 0
for i in range(len(slist)-2):
if slist[i:i+3]==['h','h','t']:
count=count+1
return count

def avgc():
sum2=0
for i in range(100):
sum2=sum2+check(100)
avg=sum2/100.0
return avg

sum1=0
for i in range(100):
sum1+=avgc()

avg2=sum1/100.0
print avg2

Answer 66

i fixed a mistake, but its pretty much the same thing

Answer 67

12.25* is your avg answer

Answer 68

for HHH and HTT and HHT

Answer 69

I read a document discussing how to find the probability of a "triplet" (for the diagram se attached file): "It is clear from the figures that one can arrive at either goal state by
many different paths involving many different numbers of steps. To find
the expected number of tosses for any given triplet, one could multiply the
length of every unique path to that triplet with its probability of occurrence
and take the sum of products over all unique paths".

Answer 70

@kingGeorge

Answer 71

i keep getting 12.25 for all the possible sequence of 3

Answer 72

maybe that wikipedia article is saying

Answer 73

The way I would approach finding the expected value of HHT, is to find the total number of length 3 sequences, and then multiply by the probability of getting HHT, or 1/8.

So if you flip the coin 100 times, there are 98 total sequences of length 3. So the expected value would be 98/8=12.25.

Answer 74

if you consider the choices
like one guy picks HHH and the other guy picks HHT
then for that game, there is a difference chance of winning

Answer 75

that is what i thought the first time around too

Answer 76

but he said this wikipedia article had something else to say about it
http://en.wikipedia.org/wiki/Penney's_game#Analysis_of_the_three-bit_game

Answer 77

i think that wikipedia article is saying something different that is how to win, based on the opponents choice or something

Answer 78

The game is completely different than what we want to solve. Penney's game is asking for the probability that a certain sequence occurs \(first\). The problem we're given has a fixed number of coin flips.

Answer 79

ya for example
if u pick HHH and I pick THH
then my 2HHs after i get a T can result from 2 of your waits
so i have a chance of finishing faster

Answer 80

"An intuitive explanation for this result, is that in any case that the sequence is not immediately the first player's choice, the chances for the first player getting their sequence-beginning, the opening two choices, are usually the chance that the second player will be getting their full sequence. So the second player will most likely "finish before" the first player.[2]"

Answer 81

ok, I understand, thank you for taking time to answer my question, really appreciate it!

Answer 82

the probabililty you see for the HHH and THH will be the same
but which one is faster is different

Answer 83

u want a program for this too xD

Answer 84

These nontransitive games are weird to think about for me. But the way I see it, every sequence is equally likely to show up at some point, but some are more likely to occur before certain other sequences. To get the nontransitivity bit, it requires you to choose two sequences in a certain order, but when we're talking about expected value, we only choose a single sequence.

Answer 85

or you can try fiddling around with that program yourself its pretty easy to implement it

Answer 86

you set up 2 counters
for each game
and award victory to the first sequence completed and repeat this process see the prob of win over 100 times and so on

Answer 87

yes, I can try it on my own, good exercise in python as well :) Thanks again!

Answer 88

sure !! see ya around

Answer 89

see ya :)