The accompanying tree diagram represents a two-stage experiment. (Let x = 0.3, y = 0.7, r = 0.55, s = 0.45, t = 0.55, and w = 0.45.)
(a) Find P(A) · P(D | A).
Correct: Your answer is correct.
(b) Find P(B) · P(D | B).
Incorrect: Your answer is incorrect.
(c) Find P(A | D).
To find P(A) · P(D | A), we need to first find the value of P(A), which represents the probability of event A occurring, and P(D | A), which represents the probability of event D occurring given that event A has occurred.
According to the tree diagram, event A occurs with a probability of x = 0.3. Therefore, P(A) = 0.3.
Now, we need to find P(D | A), which represents the probability of event D occurring given that event A has occurred. From the tree diagram, we can see that event D occurs when event A followed by event C occurs. The probability of event C occurring given that event A occurs is r = 0.55. Therefore, P(D | A) = r = 0.55.
To find P(A) · P(D | A), we simply multiply the individual probabilities:
P(A) · P(D | A) = 0.3 * 0.55 = 0.165
Therefore, the answer to part (a) is 0.165.
To find P(B) · P(D | B), we follow a similar process. From the tree diagram, we can see that event B occurs with a probability of y = 0.7. Therefore, P(B) = 0.7.
Now, we need to find P(D | B), which represents the probability of event D occurring given that event B has occurred. From the tree diagram, we can see that event D occurs when event B followed by event C occurs. The probability of event C occurring given that event B occurs is s = 0.45. Therefore, P(D | B) = s = 0.45.
To find P(B) · P(D | B), we simply multiply the individual probabilities:
P(B) · P(D | B) = 0.7 * 0.45 = 0.315
Therefore, the answer to part (b) is 0.315.
To find P(A | D), we need to find the probability of event A occurring given that event D has occurred.
Using Bayes' theorem, the formula to find P(A | D) is:
P(A | D) = (P(D | A) * P(A)) / P(D)
We have already found P(D | A) as r = 0.55 and P(A) as 0.3. Now, we need to find P(D), which represents the overall probability of event D occurring.
From the tree diagram, we can see that event D occurs when either event A followed by event C occurs or event B followed by event C occurs. These two paths have probabilities of r = 0.55 and s = 0.45, respectively.
Therefore, P(D) = P(A) * P(C | A) + P(B) * P(C | B) = (0.3 * r) + (0.7 * s) = (0.3 * 0.55) + (0.7 * 0.45).
After calculating P(D), we can substitute the values into the Bayes' theorem formula to find P(A | D):
P(A | D) = (P(D | A) * P(A)) / P(D)
Finally, calculate the result to find P(A | D).
Keep in mind that the values of r, s, x, y, t, and w that were given in the question should be used in the calculations.