Two weighted coins, determining which has a higher probability of landing heads A friend of mine asked me the following question, and I am not sure how to solve it:

You are given two weighted coins, $C_1$ and $C_2$.  Coin $C_1$ has probability $p_1$ of landing heads and $C_2$ has probability $p_2$ of landing heads.  The following experiment is preformed:
Coin $C_1$ is flipped 3 times, and lands heads 3 times.
Coin $C_2$ is flipped 10 times, and lands heads 7 times.
Based on this experiment, choose the coin which is more likely to have a higher probability of being heads.  In other words, which is more likely: $p_1&gt;p_2$ or $p_2&gt;p_1$.

Intuition tells me coin $C_1$ is the better choice, but this could be wrong, and I am wondering how do you solve this in general.  Consider the experiment, $C_1$ is flipped $n_1$ times and lands heads $m_1$ times, $C_2$ is flipped $n_2$ times and lands heads $m_2$ times.
Thanks for the help,
Edit: I think this might answer some questions:  Suppose that the probabilities of the coins, $p_1$ and $p_2$ are chosen uniformly from $[0,1]$.
As Henry mentions, I think one needs some information about the prior distributions of the weights.

Denote by $r$ the weight of a coin.  Suppose that the weight has some prior distribution $g(r)$.  Let $f(r|H=h, T=t)$ be the posterior probability density function of $r$ having observed $h$ heads and $t$ tails tossed.  Bayes' Theorem tells us that:
$$
f(r|H=h, T=t) = \frac{Pr(H=h|r, N = h+t)g(r)}{\int_0^1 Pr(H=h|r, N = h+t)g(r)\ dr}.
$$

This should allow you to answer your question.  Once you have the posterior pdf for each coin, just find their respective expected weights.
I agree with the beta approach, but given the question, I think it makes more sense to plot out the results and compare visually:

$x$ &lt;- seq($0,1,$length $= 1000$) #set $x$ from $0$ to $1$ since we're looking at probabilities
$y_1$ &lt;- dbeta$(x,4,1)$ #calculate density based on a prior of $[1,1]$ and a posterior of $[3,0]$.
$y_2$ &lt;- dbeta$(x,8,4)$ #calculate density based on a prior of $[1,1]$ and a posterior of $[7,3]$.
plot($x,y_1$,type = &quot;l&quot;) #plot density of coin 1 in black
lines($x,y2$,col $= 2$) #plot density of coin 2 in red


This yields the following:
Think of the densities as representing the &quot;probability of probability&quot;.

Coin 1 (represented by the black line) has a higher probability of having a higher probability of showing heads.

Coin 2 (represented by the red line) has a lower probability of having a higher probability of showing heads (or a higher probability of having a lower probability of showing heads...).