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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability1 Mark
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 the 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 $E_1, E_2$ and $E_3$​​​​​​​ be the events that boxes I, II and III are chosen.
$P (E_1) = P (E_2) = P (E_3)$ = $\frac{1}{3}$
let A be the event the coin drawn is of gold.
$p(A|{E_1}) = \frac{2}{2} = 1$
$P(A|{E_2}) = 0$
$P(A|{E_3}) = \frac{1}{2}$
$P({E_1}|A) = \frac{{P({E_1})P(A|{E_1})}}{{P({E_1})P(A|{E_1}) + P({E_2}) + P(A|{E_2}) + P({E_3})P(A|{E_3})}}$
$=\frac{\frac{1}{3}×1}{\frac{1}{3}×1+\frac{1}{3}×0+\frac{1}{3}×\frac{1}{2}}$
$ = \frac{2}{3}$

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