>>1141002The gold key has a 0.75 chance of working.
The silver key part follow a Binomial Distribution that can be used to work out the probability of at least 2 keys working.
Here is the proof that the silver keys are better:
Let X be the number of working keys
there is a 0.75 chance of *each* silver key working, and there are 3 silver keys.
Binomial Distributions work such that the number of trials and the probability of each trial succeeding is taken into account.
for any binomial distribution, X~(n,p) [X being distributed with n number of trials and p probability]
this is where things get a little complicated... ( i wonder if anyone will actually read this )
P(X=x) = (nCx)*(p)^x*(1-p)^(n-x) [The probability that the number of working keys "X" is equal to "x" is n Choose x (search up mathematical combinations if you dont know what that is) multiplied by the probability to the power of x, multiplied by 1-p to the power of x]
so lets put this into the context of our problem:
X~B(3,0.75) [X is Binomially Distributed with 3 being the number of trials, and 0.75 being the chance that each trial succeeds]
P(X>=2) = P(X=2) OR P(X=3) [The probability that at least 2 keys are working is the same as the probability that 2 or 3 keys are working]
P(X=2) = 3C2*(0.75)^2*(0.25)^1 = 0.421875
P(X=3) = 3C3*(0.75)^3*(0.25)^0 = 0.421875 [they are the same due to the symmetrical nature of the binomial curve]
therefore the probability that the silver keys work is 0.84375, which is more than the probability of the gold key working.
