Two sets of candidates are competing for the positions on the board of directors of a company. The probabilities that first and second sets will win are 0.6 and 0.4 respectively. If the first set wins the probability of introducing a new product is 0.8 and the corresonding probability if the second set wins is 0.3. The probability that the new product will be introduced is $0.1 k$ . Find the value of $k$ .

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Solution

Let $P (A_{1}) =$ Probability that the first set wins $= 0.6$ $P (A_{2}) =$ probability that the second set wins $= 0.4 P (B) =$ probability that a new product is introduced $∴ P (B) = P (B \cap A_{1}) + P (B \cap A_{2}) = P (A_{1}) P (B ∣ A_{1}) + P (A_{2}); P (B ∣ A_{2}) . . . (1)$
Now as per hypothesis $P (B ∣ A_{1}) = 0.8, P (B ∣ A_{2}) = 0.3$ Hence from $(1), P (B) = 0 \cdot 6 \times 0 \cdot 8 + 0 \cdot 4 \times 0 \cdot 3 = 0.60$

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