Total Probability Theorem

Total Law of Probability and Decision Trees

The total law of probability states that the probability of two or more independent events occurring is the product of their individual probabilities. Decision trees are a graphical representation of the total law of probability. They allow you to visualize the possible outcomes of a decision and the probability of each outcome.

Total Probability Theorem:

The total probability theorem states that the probability of an event occurring is the sum of the probabilities of the individual events that make up the event.

How to State and Prove Total Probability Theorem:

The total probability theorem states that the probability of an event is the sum of the probabilities of the individual outcomes that make up the event. To prove the theorem, we use the principle of addition.

The principle of addition states that the probability of two events occurring is the sum of the probabilities of the individual events occurring.

For example, the probability of flipping a coin and getting heads is 1/2, the probability of flipping a coin and getting tails is 1/2, and the probability of flipping two coins and getting two heads is 1/4.

Total Probability Theorem Proof:

The proof of the theorem is simple. We will use the fact that the probability of any event is the sum of the probabilities of the individual outcomes that make up the event.

We will start with a simple example. Suppose we have a die and we want to find the probability of getting a three. There are six possible outcomes, and only one of them is a three. So, the probability of getting a three is 1/6.

Now, let’s suppose we have a deck of cards and we want to find the probability of getting a seven. There are 52 possible outcomes, and only one of them is a seven. So, the probability of getting a seven is 1/52.

Now, let’s suppose we have a bag of marbles and we want to find the probability of getting a blue marble. There are six possible outcomes, and only one of them is blue. So, the probability of getting a blue marble is 1/6.

Now, let’s suppose we have a bag of marbles and we want to find the probability of getting a red marble. There are six possible outcomes, and two of them are red. So, the probability of getting a red marble is 2/6.

Total Probability Theorem Examples:

(1) A fair coin is flipped three times. What is the probability of getting at least two heads?

There are three possible outcomes for the first flip: heads, tails, or a tie. There are two possible outcomes for the second flip: heads or tails. Finally, there is only one possible outcome for the third flip: heads or tails. Therefore, the probability of getting at least two heads is 3/8.

(2) A jar contains ten green and ten red balls. If a ball is selected at random from the jar, what is the probability that it is red?

There are ten possible outcomes for the ball selected: nine red balls and one green ball. Therefore, the probability that the ball is red is 9/10.

about Total Probability Theorem-proof:

Let \(A\) and \(B\) be any two events in a sample space \(S\). The total probability theorem states that

\[P(A\cup B) = P(A) + P(B) – P(A\cap B)\]

Proof:

We will use mathematical induction to prove the theorem.

The base case is when \(A\) and \(B\) are mutually exclusive, meaning that \(A\cap B = \emptyset\). In this case,

\[P(A\cup B) = P(A) + P(B) – P(A\cap B) = P(A) + P(B) = 1\]

Now we will assume the theorem is true for all pairs of events \(A\) and \(B\) that are subsets of a given sample space \(S\), and show that it is also true for the pair \(A\) and \(B\) itself.

Let \(A\) and \(B\) be any two events in a sample space \(S\). We will show that

\[P(A\cup B) = P(A) + P(B) – P(A\cap B)\]

By definition,

\[P(A\cup B) = P(A) + P(B) – P(A\cap B) = P(

To understand how you can use the decision tree in calculating full potential, consider the following example:

You are considering investing in a new software system that will help your business to automate some of its operations.

The system has two options:

1) Buy the software outright for a one-time cost of $10,000

2) Rent the software for a monthly cost of $100

Which option should you choose?

The decision tree can help you to calculate the full potential of each option.

In option 1, you would pay a one-time cost of $10,000 for the software. In option 2, you would pay a monthly cost of $100 for the software.

If you use the software for 10 months, you would pay a total of $1,000 for the software in option 1, and you would pay a total of $1,200 for the software in option 2.

So, in option 2, you would pay an extra $200 for the software.

Then the chances of the second card becoming king or not will be represented by the law of full probabilities such as:

P(K|H) = 1/2

P(K|H) = 1/2

Explanation of Total Probability Theorem with the Following Examples:

Example 1:

In a jar there are 5 red balls and 10 blue balls. What is the probability that a randomly chosen ball is blue?

There are 10 blue balls and 15 balls in total. So, the probability of choosing a blue ball is 10/15.