Multiple choice

Die A has $4$ red and $2$ white faces where as die $B$ has two red and $4$ white faces. $A$ fair coin is tossed. If head turns up, the game continues by throwing die A, if tail turns up then die B is to be used. Probability of getting a red face at any throw is

  1. $\displaystyle \frac{1}{2}$
  2. $\displaystyle \frac{2}{3}$
  3. $ \displaystyle\frac{1}{3}$

  4. none of these
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A Correct answer
Explanation

P(Red) = P(Red|A)P(A) + P(Red|B)P(B). P(A) = 1/2, P(B) = 1/2. P(Red|A) = 4/6 = 2/3. P(Red|B) = 2/6 = 1/3. P(Red) = (2/3 * 1/2) + (1/3 * 1/2) = 1/3 + 1/6 = 1/2.

AI explanation

By the total probability theorem, the probability of getting a red face is P(Head) * P(Red from A) + P(Tail) * P(Red from B). Substituting the values gives (1/2) * (4/6) + (1/2) * (2/6). This calculates to 4/12 + 2/12 = 6/12 = 1/2.

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