Die A has four red and two white faces whereas die B has two red and four white faces. A coin is flipped once. If it falls a head, the game continues by throwing die A, if it falls tail then die B is

Question

Die A has four red and two white faces whereas die B has two red and four white faces. A coin is flipped once. If it falls a head, the game continues by throwing die A, if it falls tail then die B is to be used. If the probability that die A used is 33 32 ​ when it is given that red turns up every time in first n throws, then the value of n is ____ 。

TestHub · Accepted Answer

Let R be the event that a red face appears in each of the first n throws.

$E_{1}$ : Die A is used when head has already fallen

$E_{2}$ : Die B is used when tail has already fallen.

$∴ P (R / E_{1}) = (\frac{2}{3})^{n}$ and $P (\frac{R}{E _{2}}) = (\frac{1}{3})^{n}$

As per the given condition

$\frac{P ( E _{1} ) \cdot P ( R / E _{1} )}{P ( E _{1} ) \cdot P ( R / E _{1} ) + P ( E _{2} ) \cdot P ( R / E _{2} )} = \frac{32}{33}$

$\Rightarrow \frac{1/2 \cdot ( 2/3 ) ^{n}}{\frac{1}{2} \cdot ( \frac{2}{3} ) ^{n} + \frac{1}{2} ( \frac{1}{3} ) ^{n}} = \frac{32}{33} \Rightarrow \frac{2 ^{n}}{2 ^{n} + 1} = \frac{32}{33}$

$\Rightarrow n = 5$

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