322 consecutive "heads": formula for calculating?

Does anyone have a formula for how I can calculate the probability of 322 consecutive heads in a coin-toss game?

let h be the probability of a head
let t be the probability of a tail
[for a true coin both probability's are 1/2]

[h+t]=1
for n tosses we have
[h+t]^n=1^n
by the binomial expansion
h^n+nh^(n-1)t /1! +.... t^n =1

probibility of all heads for n=322:
h^322 answer
if h=1/2
322 heads in 322 tosses = 1/2^322

probability of 321 heads and 1 tail
322h^321 t^1 /1!

etcetera

Arthur

Re: 322 consecutive "heads"

bobW said:

Does anyone have a formula for how I can calculate the probability of 322 consecutive heads in a coin toss game.

Click to expand...

Is “a coin toss game” a particular game?
If we flip any coin n times the probability of getting n heads is \(\displaystyle \left( {\frac{1}{2}} \right)^n\).
If we flip it 10 times the probability of getting the string HHHHHHHHHT is \(\displaystyle \left( {\frac{1}{2}} \right)^{10}\).
In fact that is the probability of getting any particular string of ten flips.

the probability of any specific sequence of 10 tosses is [1/2]^10 for a "good" coin
for instance hhhhhhhhht

this is different from ," what is the probability of only 1 tail and 9 heads in 10 tosses?"
then you have
hhhhhhhhht or hhhhhhhhth or hhhhhhht hh etc 10 [1/2]^10

[h+t]^10=
h^10/0! ten heads
10 h^9t/1! nine heads 1 tail
10[9]h^8t^2/2! eight heads 2 tails
etcetera down to
t^10 ten tails

Arthur

arthur ohlsten said:

the probability of any specific sequence of 10 tosses is [1/2]^10 for a "good" coin
for instance hhhhhhhhht

this is different from ," what is the probability of only 1 tail and 9 heads in 10 tosses?"

Click to expand...

Actually the myth of the “a good coin” was exploded by two physicists at Cal Tech (I think) about 10 years ago. They showed if one “flips” a coin, send spinning and catch it on the fly, then there is no way to bias a coin unless it has two identical sides. It is a easy though experiment: as it spins in the air weight has nothing to do with its spin.

Moreover, I said nothing about there being one tail and n consecutive heads. And neither does the problem. From the statement it sounds as if this is a waiting time problem until the first success, in this case a tail. What is the probability of 323 flips until the first tail.

the cal tech conclusion is true if you catch it in the air. If you let it land, then I believe you can have a h=7/8 and t=1/8 as a example. I am not sure of this as I never tested the theory, but it is true if you use a random number generator rather than tossing a coin, for the coin test.

Arthur

Re: 322 consecutive "heads"

bobW said:

Does anyone have a formula for how I can calculate the probability of 322 consecutive heads in a coin toss game.

Click to expand...

322 ?! Are you serious?
Did you know that 2^322 = a 97digit number?

and if you let in land

It can roll-of under the couch and never know what happened (quantum cat)

or it can land on mud on the edge and stay there

possibilities ...possibilities.....