# Set theory and baye's theorem problem

• bdh2991
In summary, the question is to find the probability of P(A2 U A3 | A1). Based on the given information, the probability of A2 U A3 is 0.4675 and the equation P((A2 U A3) \bigcap A1) / P(A1) can be used to find the rest of the answer. However, if this equation does not give the correct answer, another option is to use the formula P(A2 U A3|A1) = P(A2|A1) + P(A3|A1) - P(A2,A3|A1), which can be rearranged using Bayes theorem to get [ P(A1,A2) + P

#### bdh2991

A computer consulting firm presently has bids out on three projects. Let Ai = {awarded project i}, for i = 1, 2, 3, and suppose that P(A1) = 0.22, P(A2) = 0.25, P(A3) = 0.29, P(A1 ∩ A2) = 0.07, P(A1 ∩ A3) = 0.09, P(A2 ∩ A3) = 0.05, P(A1 ∩ A2 ∩ A3) = 0.02.

The question is to find the probability of P(A2 U A3 | A1)

I found that A2 U A3 is equal to 0.4675 but I'm confused on which equation to use to get the rest of the answer...

I originally thought you would just do P((A2 U A3) $\bigcap$ A1) / P(A1) but I'm not getting the correct answer...any help?

There are 8 cases, you can calculate the probabilities of all of them (or at least the 4 with A1 in it) and then just calculate P(A2 U A3 | A1) based on all cases with A1.

Try this:

P(A2 U A3|A1) = P(A2|A1) + P(A3|A1) - P(A2,A3|A1)

and you can switch around each term using Bayes theorem to get

= [ P(A1,A2) + P(A1,A3) - P(A1,A2,A3) ] / P(A1)

if I am not mistaken

edit: Oh, I just realized that is what you said you tried. Looks like it should work to me...

bdh2991 said:
I originally thought you would just do P((A2 U A3) $\bigcap$ A1) / P(A1) but I'm not getting the correct answer...any help?

## 1. What is set theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects or elements. It provides a foundation for understanding mathematical concepts such as numbers, functions, and relations.

## 2. How is set theory used in real life?

Set theory is used in a variety of fields, such as computer science, statistics, and economics. It helps to organize and classify data, make predictions and decisions, and solve problems related to sets and their elements.

## 3. What is Bayes' theorem and how does it relate to set theory?

Bayes' theorem is a mathematical formula that describes the probability of an event based on prior knowledge or information. It is often used in conjunction with set theory to calculate the likelihood of an event occurring based on the elements in a set.

## 4. How do you solve a Bayes' theorem problem?

To solve a Bayes' theorem problem, you need to first identify the prior probability (the probability of an event occurring before any new information is taken into account), the likelihood (the probability of the new information given the event), and the marginal probability (the probability of the new information occurring). Then, plug these values into the formula: P(A|B) = P(B|A) * P(A) / P(B).

## 5. What are some common misconceptions about set theory and Bayes' theorem?

One common misconception is that set theory and Bayes' theorem are only applicable in mathematics and have no real-life applications. In reality, they are used extensively in various fields, as mentioned before. Another misconception is that Bayes' theorem is a simple formula, when in fact it can be quite complex and requires careful consideration of the given information.