Boolean variables A,B,C, and D are all independent of each other. P(a) = 0.22, P(b) = 0.47, P(c)= 0.82, and P(d) = 0.59. Given this, find the following probabilities.
$ P((A \ \lor \sim C) \to (B \land D)) $
I got stuck with this I'm not sure how to simplify this?
\begin{align*} &\bigl(a \lor \lnot c) \to (b \land d)\bigr)\\[4pt] \equiv\;&\lnot (a \lor \lnot c) \lor (b \land d)\qquad\text{[since $p \to q \equiv \lnot p \lor q$]}\\[4pt] \equiv\;&(\lnot a \land c) \lor (b \land d) \qquad\;\;\,\text{[by DeMorgan's law]}\\[4pt] &\;\;\text{hence}\\[4pt] \;\;&P(\bigl(a \lor \lnot c) \to (b \land d)\bigr)\\[4pt] =\;&P\bigl((\lnot a \land c) \lor (b \land d)\bigr)\\[4pt] =\;& P(\lnot a \land c) +P(b \land d) -P\bigl((\lnot a \land c) \land (b \land d)\bigr)\\[4pt] =\;& P(\lnot a \land c) +P(b \land d) -P(\lnot a \land c \land b \land d)\\[4pt] \end{align*} But it's given that $a,b,c,d$ are independent.
Can you finish it?