Common Forecast

Let \(A\) and \(B\) mean that the respective performances run, and let \(W\) denote the shared weather condition. Conditioning on the common cause and using the stipulated conditional independence,

\(\Pr(A\cap B)=0.6(0.9)(0.9)+0.4(0.3)(0.3)=0.486+0.036=\boxed{0.522}\).

Either performance separately runs with probability \(\Pr(A)=\Pr(B)=0.6(0.9)+0.4(0.3)=0.54+0.12=\boxed{0.66}\). If they were independent without conditioning, the joint probability would be \(0.66^2=0.4356\), which is not \(0.522\); the difference, \(0.0864\), is their covariance as binary indicators. Shared good-weather days boost both events simultaneously, while shared bad-weather days suppress both.

As a full-table check, the probabilities of (both run, only A runs, only B runs, neither runs) are \((0.522,0.138,0.138,0.202)\). They are nonnegative, sum to one, and each performance's marginal success probability is \(0.522+0.138=0.66\). Conditional independence within weather categories does not imply unconditional independence after mixing over weather.