Three cooks, A, B, and C, bake a special kind of cake, and with respective proba

Question

Three cooks, A, B, and C, bake a special kind of cake, and with respective probabilities 02, 03, and 05, it fails to rise. In the restaurant where they work, cook A bakes 50 percent of these cakes, B bakes 30 percent, and C bakes 20 percent. (a) What is the probability that a randomly chosen cake will rise? Answer: (b) Given that a randomly chosen cake fails to rise, what is the conditional probability that it was backed by cook C? Answer: Consider the experiment of throwing two fair dice. Let A be the event that the sum of the two dice is 8, and let B be the event of rolling doubles. Are events A and B independent? Justify your answer.

Explanation / Answer

5. For rolling of two dice,

Total number of possible outcomes = 6*6 = 36

Number of possible outcomes with sum 8 = 5 [(6, 2), (2, 6), (5, 3), (3, 5), (4, 4)]

Hence,

P(Sum is 8) = 5/36

Number of possible outcomes with doubles = 6 [(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)]

Hence,

P(Rolling Double) = 6/36 = 1/6

Now,

P(Sum is 8 and rolling double) = 1/36 [Only one outcome satisfying this which is (4, 4)]

If two events are independent then P(A and B) = P(A) P(B)

Here,

P(Sum is 8) * P(Rolling Double)

= 5/36 * 1/6

= 5/216

Which is not equal to P(Sum is 8 and rolling double)

Hence,

These events are not independent.

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