Consider two urns: urn-1 contains 3 red, 1 black, and 3 white balls while urn-2 contains 2 red, 3 black, and 3 white balls. If an urn is randomly selected with equal chances of it being Urn-1 or Urn-2, and then a ball is picked at random from the selected urn, find the probability that either a white or a black ball is picked.
My approach to this question looks like this: $P(U1)*P(W|U1) + P(U1)*P(B|U1) + P(U2)*P(W|U2) + P(U2)*P(B|U2) =$ Probability that the ball picked is either black or white.
Does this seem correct?
Your derivation is correct, but it may be simplified by replacing black and white balls with grey balls, since at no point in the question do black and white get treated separately. So urn $1$ has $3$ red and $4$ grey balls, and urn $2$ has $2$ red and $6$ grey balls. $$P=\frac12\left(\frac47+\frac68\right)=\frac{37}{56}$$