I've been trying to figure out this question using the recursive method.
What is the expected value of number of flips to get 4 coins to all
land on heads, where once a coin lands on heads you dont have the
reflip it, Also, if you had a wand that could flip any pair of identically facing coins with unlimited uses what would the new expected value be?
For the first question, my attempt is as follows:
For one coin case, expected flip is 2, which is clear.
For 2 coins, you flip them first. There are 3 possible scenarios: 1/4 chance of having 0 head, 1/2 chance of having 1 head, 1/4 chance of having 2 heads.
If we set $a$ as the expected number of tosses needed for 2 coins, then we could set the equation as this:
$a = 1 + (1/4)*a + (1/2)*2 + (1/4) * 0 $
so that $a$ would be 8/3.
However, my intuition tells me that: by using simpler method: if we toss 2 coins, that expected number of head is 1, assuming the fair coin. And then since 1 coin is left, we add 2 since expected tosses for getting head with one coin is 2. So the total expected number would be $1 + 2
= 3$
So I'm confused which approach is right, and if one is wrong, why would it be?
Best Answer
I am not completely sure about my understanding of the question (please check).
My understanding described with $n$ replacing $4$ to make things more general:
Let there be $n$ coins. The first flip is a flip of all coins and if $k$ head appears then we put these coins aside so that the next flip will be a flip of $n-k$ coins. The process ends if all coins are put aside.
To be found is then $\mu_4$.
This can be done on base of the following equalities:
This is based on: $$\mu_n=\sum_{k=0}^nP(n-k\text{ heads appear by throwing }n\text{ coins})(1+\mu_k)$$where $\mu_0:=0$.
It is handsome to start with the last one and find $\mu_1,\mu_2,\mu_3,\mu_4$ respectively.
For the second question I have no answer mainly because I do not understand what they mean (what exactly is a "wand"?).
Edit on second problem:
The wand (a new word for me) reduces things to the question: is the number of tails that appear even or odd? Then for every $n\geq1$ we have the equality:$$\mu_n=1+\frac12\mu_1$$
Consequently $\mu_n=2$ for every $n$.