I have a problem using the correlation in combination with the "at least one" probability.
I have $P(A)=57\%$, and $P(B)=74\%$, and I calculated their correlation coefficient and it is $0.1557$.
To calculate the $P(\text{at least one})$ I did it this way:
$$P(\text{at least one happens}) = 1 - P(\text{neither A nor B happens})$$
for which I did $P(A)' = 43\%$, $P(B)' = 26\%$,
$$P(\text{neither A nor B}) = P(A)'P(B)' = (0.43)(0.26) = 0.1118.$$
So using the first formula: $$P(\text{at least one happens}) = 1 - 0.1118 = 0.8882.$$
So the probability of at least one of them happening is $88.82\%$, however, I need to combine this with their correlation and I'm having troubles doing it that way, I would appreciate it if someone could help me with this problem.
Thank you!
PD. Forgot to add their are dependent events.
What does the correlation coefficient of two events mean? Is it the correlation coefficient of their Bernoulli indicator random variables? Then
$$\newcommand{\Chi}{{\raise{0.5ex}{\chi}}} \begin{align} \rho_{\!\lower{0.5ex}{A,B}} & = {\sf Corr}(\Chi_A, \Chi_B) \\[1ex] & = \dfrac{{\sf Cov}(\Chi_A, \Chi_B)}{\sqrt{\,{\sf Var}(\Chi_A){\sf Var}(\Chi_B)\,}} \\[1ex] & = \dfrac{{\sf E}(\Chi_A\Chi_B)-{\sf E}(\Chi_A){\sf E}(\Chi_B)}{\sqrt{\,({\sf E}({\Chi}_A^2)-{\sf E}({\Chi}_A)^2)\,({\sf E}({\Chi}_B^2)-{\sf E}({\Chi}_B)^2)\;}} \\[1ex] & = \dfrac{{\sf P}(A\cap B)-{\sf P}(A){\sf P}(B)}{\sqrt{\,({\sf P}(A)-{\sf P}(A)^2)\,({\sf P}(B)-{\sf P}(B)^2)\;}} \\[2ex] \therefore {\sf P}(A\cap B) & = {\sf P}(A){\sf P}(B)+ \rho_{\!\lower{0.5ex}{A,B}} \sqrt{\,{\sf P}(A)(1-{\sf P}(A))\,{\sf P}(B)(1-{\sf P}(B))\;} \end{align} $$