This is a question aimed at people newer to statistics/probabilities. I made this question as a test to myself because I haven't done stats in AGES.
A fair coin is thrown fairly 10, 20, 50, and 100 times.
What is the probability that the same side of the coin will not occur twice in a row.
Give the probability for each one.
(Or, if you're lazy, simply give a formula for finding the probabilities
EDIT: In answer to people's questions, this is to find out what people come up with, and maybe help somebody new to statistics who wants an easy question.
So you already know the answer and just want to see what people come up with? I'll put my answer in a spoiler box then.
(Edited because my first reply was not well thought out. Same answer though, as it turns out.)Spoiler:
Edit 2: whoops. Meant to edit my first post rather than make a new post. Browser's back button didn't behave as I expected.
Here I will put the "results" a.k.a if people get it right or not.
ANDS! - Incorrect but close, although it was rather vague (didn't give a formula for finding )
undefined - Both your answers are correct
Plato - Correct =D
What Plato said applied into a formula:
Spoiler:
undefined's formulas:
Spoiler:
Also ANDS!, here's my explanation for why your answer is not correct:
Spoiler:
What I said was:
I'll try and put it another way.Instead of , it is either , or - if that makes sense.
This is because out of all the possible permutations (n amount), there are correct ones, not .
For instance, like Plato said, if there were 10 tosses, HTHTHTHTHT and THTHTHTHTH would both be correct permutations that fit the given pattern (The same side not coming up more than once in a row)
You put the formula as , where is the number of possible permutations, however the correct formula would be , because out of all the possible permutations, there are 2 that are correct.
So:
p = 2/n
However we can change that so it is
p = 1/(n/2)
Treat the fraction as an equation, so for instance it would be
2 = n
1 = n/2
See?
Note: If you don't believe me, here's an example of why 2/n = 1/(n/2)
If n = 4, then:
2/4 = 1/(4/2)
2/4 = 1/2
It should be noted that there is incorrect usage of the word "permutations" happening on this thread. A permutation refers to ways in which some elements can be re-arranged while not changing the elements themselves.
Thus THTH is a permutation of HTHT and TTHH, but not of TTTT.
Typically permutations are over a set of distinct elements, such as {a, b, c, d}, for which the number of permutations is .
Sometimes permutations are over a set of not necessarily distinct elements, i.e., a multiset, such as {a,a,b,b,b,c,c,c,c,d,d}, for which the number of permutations is .
Anyway, we are not talking about permutations but about possible sequences of coin flips.
What I mean to convey is that the word "permutation" is already in use, and means something else. Of course we can define any word to mean anything we like, but this can cause confusion because some people will think the word means what is written in the dictionary, rather than what you say it means.