Question

(1 point) Rework problem 1 from section 3.4 of your text, involving probabilities on a tree diagram. Construct a copy of figure 3.15 in your text, where the first outcome is one of A and B the second outcome in each case is one of a and b. Use the following probabilities instead of those given in your text: Pr[A]= 2/5 Pr[B]= 3/5 Pr[a∣A]= 8/9 Pr[b∣A]= 1/9 Pr[a∣B]= 5/7 Pr[b∣B]= 2/7 Find the following missing probabilities: (1) Pr[B]= (2) Pr[b]= (3) Pr[b∣B]= (4) Pr[B∣b]=

Answer 1

To solve the student's question, we use the provided conditional probabilities and the law of total probability for Pr[b], and Bayes' theorem for Pr[B|b].

Step-by-step explanation:

The student is tasked with calculating probabilities using a tree diagram based on provided conditional probabilities. The first part of the tree has outcomes A and B with respective probabilities Pr[A]= 2/5 and Pr[B]= 3/5.

For each of these, there are further outcomes a and b with probabilities conditional on the first outcome. They are given as follows: Pr[a|A]= 8/9, Pr[b|A]= 1/9, Pr[a|B]= 5/7, and Pr[b|B]= 2/7. To find the missing probabilities:

1. Pr[B] is already provided as 3/5.
2. Pr[b] can be found using the law of total probability: Pr[b] = Pr[b|A]Pr[A] + Pr[b|B]Pr[B] = (1/9)(2/5) + (2/7)(3/5).
3. Pr[b|B] is provided as 2/7.
4. To find Pr[B|b], we apply Bayes' theorem: Pr[B|b] = Pr[b|B]Pr[B] / Pr[b].