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 $Y$?

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.
