The probability of drawing a one-hundred rupee currency from two boxes one of which contain 3 fifty-rupee currency and 2 one-hundred rupee and other box contain 2 fifty-rupee currency and 3 one-hundred rupee currency notes is-
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{2} \)
A box is chosen at random and a note is drawn from it, so each box has probability \( \frac{1}{2} \). Let \( E_{1},E_{2} \) be the events of choosing Box 1 and Box 2, and H the event of drawing a one-hundred rupee note:
\( \displaystyle P(E_{1})=P(E_{2})=\frac{1}{2} \)
\( \displaystyle P(H\,|\,E_{1})=\frac{2}{5} \)
\( \displaystyle P(H\,|\,E_{2})=\frac{3}{5} \)
By the theorem of total probability:
\( \displaystyle P(H)=P(E_{1})P(H\,|\,E_{1})+P(E_{2})P(H\,|\,E_{2}) \)
\( \displaystyle =\frac{1}{2}\cdot\frac{2}{5}+\frac{1}{2}\cdot\frac{3}{5} \)
\( \displaystyle =\frac{1}{2} \)