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Probability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSProbability5 Marks
Question
Given three identical boxes $I, II$ and $III$ each containing two coins. In box $I,$ both coins are gold coins, in box $II,$ both are silver coins and in box $III,$ there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?
✓

Answer

Let $E_1:$ selecting box $I, E_2:$ selecting box $II$ and $E_3:$ selecting box $III$
$ \therefore P (E_1) = P (E_2) = P (E_3) = \frac{1}{3}$ let event $A:$ Getting a gold coin
$\therefore P (A/E_1) = 1 P (A/E_2) = 0 P (A/E_3)$
$= \frac{1}{2} P(E_1/A)$
$= \frac{\text{P(E}_{1})\cdot\text{P(A/E}_{1})}{\text{P(E}_{1})\text{P(A/E}_{1})+\text{P(E}_{2})\text{P(A/E}_{2})+\text{P(E}_{3})\text{P(A/E}_{3})}$
$= \frac{\frac{1}{3}\cdot1}{\frac{1}{3}\cdot1+0+\frac{1}{3}\cdot\frac{1}{2}}$
$=\frac{2}{3}.$

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