combinatoric problem related to drug i want to choose optimal decision from following problem
Imagine having been bitten by an exotic, poisonous snake. Suppose the ER
physician estimates that the probability you will die is $1/3$ unless you receive
effective treatment immediately. At the moment, she can offer you a choice of
experimental antivenins from two competing ‘‘snake farms.’’ Antivenin $X$ has
been administered to ten previous victims of the same type of snake bite and
nine of them survived. Antivenin $Y$, on the other hand, has only been
administered to four previous patients, but all of them survived. Unfortunately,
mixing the two drugs in your body would create a toxic substance much
deadlier than the venom from the snake. Under these circumstances, which
antivenin would you choose, and why?
so first off all i have concluded that, for  substance $X$,i would have  $90$%  chance to be survivded,and for $Y$ ,i would have $100$%,so maybe it should be indicator for me to choose $Y$,on the other hand,if we consider it as a combinatoric problem,then we have $p=2/3$ if we don't die when get medical treatment, and $q=1/3$ if we do,so for substance $X$,we would have
  $(10!/9!)*((p)^{10})*q=0.05202459$
while for substance $Y$,it would be
   $4!/4!*(2/3)^4*(1/3)^0=16/84=0.19047619$
so it means that i have  more chance for $Y$,so  does it means that i should choose $Y$?
 A: This question cannot be answered without knowledge of your prior assessment of the effectiveness of the antivenins. If X was produced by a company that usually produces effective drugs and Y was produced by a quack, the result will be different than if it was the other way around. If you don't have any prior information on the likely effectiveness of the antivenins, you'll need to make some assumption that will be to some degree arbitrary. For instance, since you'd survive with probability $2/3$ without the antivenin, you could assume a uniform distribution between $2/3$ and $1$ for the chance of surviving after taking the antivenin, for each of the two antivenins (assuming you're confident that they're not harmful). Then with $j$ trials successful and $k$ trials unsuccessful for antivenin $i$, the posterior distribution for the survival probability $p_i$ of antivenin $i$ would be
$$\frac{p_i^j(1-p_i)^k}{\int_{2/3}^1p^j(1-p)^k\mathrm dp}\;,$$
and your survival probability if you take antivenin $i$ would be
$$\int_{2/3}^1\frac{p_i^j(1-p_i)^k}{\int_{2/3}^1p^j(1-p)^k\mathrm dp}p_i\mathrm dp_i\;,$$
which comes out as $502769/589806\approx0.852$ for X and $3325/3798\approx0.875$ for Y.
A: To follow up on joriki's answer...
Let's consider the case that you don't make the assumption that the antivenins aren't harmful, but rather just start from the assumption that any survival rate between 0 and 1 is equally likely for a given antivenin a priori, before any trials have been carried out.
Then the expected survival rate for an antivenin, after $j$ successful and $k$ unsuccessful trials have been carried out, is
$$\int_0^1\frac{p^j(1-p)^k}{\int_0^1q^j(1-q)^k\mathrm dq}p\,\mathrm dp\,,$$
which works out to simply
$$\frac{j+1}{j+k+2}.$$
Thus, under these assumptions, you should expect antivenin X to give you a
$$\frac{9+1}{9+1+2} = \frac{10}{12} = \frac{5}{6} \approx 83\%$$
chance of survival, while the corresponding probability for antivenin Y should also work out to
$$\frac{4+1}{4+0+2} = \frac{5}{6} \approx 83\%.$$
Thus, under these assumption, it doesn't matter which antivenin you take — either of them will cut your chance of dying down by half, from $\frac13$ down to $\frac16$.
This simple "rule of thumb", that after $x$ successful and $y$ unsuccessful trials you should expect the success rate to be $\frac{x+1}{x+y+2}$, is a very old and classic principle known as the rule of succession.  It was introduced by Pierre-Simon Laplace, who famously used it to calculate the probability that the sun will rise tomorrow, given that it has done so every day for at least the past 5000 years.  It may also well be what the author of your textbook was expecting you to apply.
(Of course, if you did assume a uniform prior between 0 and 1 for the survival rate after taking an untested antivenin, then you should never agree to be the first person to try one, since the $\frac12$ expected survival rate predicted by the rule of succession is less than the $\frac23$ postulated for taking no treatment.  Thus, the first person who tried each antivenin must either have assumed a different prior, or they must've been very stupid or very altruistic.)
