Three bags contain a number of red and white balls as follows:
Bag I: 3 red balls
Bag II: 2 red balls and 1 white ball
Bag III: 3 white balls
The probability that bag \( i \) will be chosen, and a ball is selected from it is \( \displaystyle\frac{i}{6},i=1,2,3 \). The probability that a red ball is selected is equal to:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 7/18
The bags are chosen with probabilities
\( \displaystyle P(E_1)=\frac{1}{6} \)
\( \displaystyle P(E_2)=\frac{2}{6} \)
\( \displaystyle P(E_3)=\frac{3}{6} \)
and the chance of a red ball from each bag is
\( \displaystyle P(R|E_1)=\frac{3}{3}=1 \)
\( \displaystyle P(R|E_2)=\frac{2}{3} \)
\( \displaystyle P(R|E_3)=0 \)
By the theorem of total probability,
\( \displaystyle P(R)=\frac{1}{6}(1)+\frac{2}{6}\cdot\frac{2}{3}+\frac{3}{6}(0) \)
\( \displaystyle =\frac{1}{6}+\frac{2}{9} \)
\( \displaystyle =\frac{3}{18}+\frac{4}{18} \)
\( \displaystyle =\frac{7}{18} \)