Conditional probability

Lesson 6 Chapter 2

One of the essential concepts in probability theory is the **conditional probability**. This is because a lot of events **depends** on ** other precedent events** or

**. Recognizing and calculating this dependency can lead to a more precise probability estimation. As below figure,**

*available partial information***!!**

*how many layers should you wear, depends on the weather***Conditional probability definition**

Let's have an example. Assume we toss a **dice** and then flip a fair **coin**. What is the probability of getting number one (Event ) in tossing the dice **and** getting heads (Event ) after flipping the coin? Clearly . So, the probability of getting number one (in tossing the dice) **and** getting heads (in flipping the coin) is as below:

Let's take a look at a conditional situation. Assume we want to calculate the probability of *having heads (in flipping the coin) **if** we get one (in tossing the dice)*? This is a **conditional statement**. Basically, is conditioned on happening :

The mathematical formulation of the conditional probability of two events is as below:

**Conditional probability calculation**

**Question**

Solution: Let's denote events as follows: **(A)** summation of the two numbers is higher than eight, and **(B)** we get the number 4 in our first trial**. **Definitely, and are not independent, and we want to calculate . The calculation of is easy. As we may see, all numbers with equal probability in tossing a dice, then . Now we need to calculate . The event indicates that we already know that the first dice number was 4. What is the event space of ? In other words, what should we see as the number of the second dice that makes the summation higher than 8? The following pairs demonstrate the event space of as the first and second elements of the pairs are associated with the first and second tossing of the dice:

The sample space is as below:

So the event space has only two elements, and the sample space has 36 elements. So . So the final answer would be as follows: