Question

Three local textile distributors (K, L, and M) are competing for a contract to supply textiles to Company XYZ. The probabilities that textile distributors K, L, and M will win the contract are 0.4, 0.3, and 0.3, respectively. If textile distributors K, L, and M win the contract, the probabilities that they will make profits are 0.65, 0.85, and 0.45, respectively. a) Draw a tree diagram for the above information. b) Calculate the probability that the contract was awarded to textile distributor K given that the contract is found to be unprofitable.

Answer 1

(a) A tree diagram that shows the probabilities of each event is as follows:
[asy]
unitsize(0.6cm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
label("Company XYZ",(0,0));
label("K",(2,-1),SE);
label("L",(0,-1),SW);
label("M",(-2,-1),SW);
label("$0.4$",(2,-1));
label("$0.3$",(0,-1));
label("$0.3$",(-2,-1));
draw((0,0)--(-2,-2)--(-1,-3),Arrows);
draw((0,0)--(0,-2)--(0,-3),Arrows);
draw((0,0)--(2,-2)--(1,-3),Arrows);
label("$0.65$",(2,-2));
label("$0.85$",(0,-2));
label("$0.45$",(-2,-2));
label("Profit",(2,-3));
label("No profit",(0,-3));
label("Profit",(-2,-3));
[/asy] (b) The probability of the contract going to textile distributor K given that the contract is found to be unprofitable is asked to be calculated.Probability of K winning the contract and making no profit is: P(K and no profit) = P(K) * P(no profit | K) = 0.4 * (1 - 0.65) = 0.14Probability of L winning the contract and making no profit is: P(L and no profit) = P(L) * P(no profit | L) = 0.3 * (1 - 0.85) = 0.045Probability of M winning the contract and making no profit is: P(M and no profit) = P(M) * P(no profit | M) = 0.3 * (1 - 0.45) = 0.165The probability of no profit is P(K and no profit) + P(L and no profit) + P(M and no profit) = 0.14 + 0.045 + 0.165 = 0.35The probability of K winning the contract given no profit is: P(K | no profit) = P(K and no profit) / P(no profit)= 0.14/0.35= 0.4. Answer: The probability that the contract was awarded to textile distributor K given that the contract is found to be unprofitable is 0.4.