Conditional Probability

In this section, we'll talk about conditional probability, a common topic in probability.

What is Probability?

Finding the probability of something is the same as finding the chance that something will happen. For example, suppose you roll a dice. What is the probability that you roll a 6? Well, since there are 6 choices, and each has an equal chance of being spun, the probability of getting exactly a 6 is 1/6.

Independent
Events

 

Every event has some probability of happening. Independent events are events that do not impact the chance of the other happening. For example, if I flip a dice twice, the outcome of the first flip does not impact the outcome of the second flip. 

What does this mean from a probability standpoint? This means that the probability of one event does not change the probability of the other. In other words, the probabilities are constant.

Going back to the dice example, the probability of getting a 5 on the first flip is 1/6. But the probability of getting a 5 on the second flip is still 1/6.

Dependent 
Events

Dependent events are completely opposite of independent events. Two events are dependent if the outcome of one event depends on the other. For example, suppose we have a bag with 2 red cards and 3 black cards. Suppose we draw 2 random cards from the bag. Then the probability of the second card being red is dependent on the color we draw on the first turn.

What does this mean form a probability standpoint? This means that the probability of one event changes the probability of the second event. In other words, the probability of the second event is not constant

Conditional 
Probability

Conditional Probability follows a similar sentiment to our discussion of dependent events. Conditional Probability refers to the probability of one event happening based on the occurrence of another event happening. For example, suppose there is a 30% chance of rain. Then this event will change the probability of bringing an umbrella. As a more concrete example, consider the following:

A fair die has been rolled and we want to find the probability that it is a 6. The answer is clearly 1/6. But suppose we have extra information that the number rolled was even. Since there are 3 possible ways to roll an even number, each being equally likely, our probability changes to 1/3. The revised probability that event A has occurred, given that another event B has definitely occurred, is called the conditional probability of A given B and is denoted by P(A|B). 

In following sections, we'll show how to mathematically deduce these probabilities.

Exercises:

Classify the following as independent or dependent

1. The amount of hard work, and your grades in school

2. Rolling a 6 on a dice, rolling a 6 again

3. Flipping a coin twice, getting heads the first time, tails the second time

4. Normal Deck of Cards: Drawing a red card, the drawing a black card.

5. Reading Speed, time to finish reading a book.