Going back to Pascal’s wager, let
X = 0 represent disbelief when a benevolent God doesn’t exist
X = 1 represent disbelief when a benevolent God does exist
Y = 2 represent belief when a benevolent God does exist
Y = 3 represent belief when a benevolent God does not exist
Presume that p is the likelihood that a benevolent God exists. Then you can compute the expected value of disbelief and the expect value of belief by first creating a value function. Below, for argument sake we are somewhat randomly assign a value of one million to disbelief if a benevolent God doesn’t exist. The conclusions are the same if you choose any other finite number...
\begin{gather*}
v(0) = 1,000,000, f(0) = 1-p\\
v(1) = -\infty, f(1) = p\\
v(2) = \infty, f(2) = p\\
v(3) = 0, f(3) = 1-p
\end{gather*}
Then, looking at the disbelief random variable X with a probability function \(f(0)=1-p, f(1)=p\text{,}\)
\begin{align*}
E[\text{disbelief}] & = v(0)f(0) + v(1)f(1)\\
& = 1000000 \times (1-p) - \infty \times p\\
& = -\infty
\end{align*}
if p>0. On the other hand, looking at the belief random variable Y with probability function \(g(2)=p, g(3)=1-p\text{,}\)
\begin{align*}
E[\text{belief}] & = v(2)g(2) + v(3)g(3)\\
& = \infty \times p + 0 \times (1-p)\\
& = \infty
\end{align*}
if p>0. So Pascal’s conclusion is that if there is even the slightest chance that a benevolent God exists then belief is the smart and scientific choice.