CH3 - Independent Events

This chapter will cover the concept of independent events in probability theory. Understanding independent events is crucial for various statistical analyses and decision-making processes. We will explore the definition of independent events, their properties, and how to calculate probabilities involving independent events.

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Independent Events

Say that we have 2 sequential events $A,B$. The outcome of the first event does not effect the likelihood of the second event occurring.

We distinguish independent events with things like rolling a die once/twice, or flipping a coin a certain amount of times.

No matter what, the outcome of a previous trial of these tests will never effect the future trials.

On the other hand, we have dependent events, like picking a card after already taking one out.

Example

What is the probability of pulling an ace out of a deck of cards?

Solution

The probability of pulling an ace out of a deck of cards is $\frac{4}{52}=\frac{1}{13}$.

This makes perfect sense, as there are 4 aces in a deck of 52 cards.

Example

Continuing from the previous example, what is the probability of pulling another ace after already pulling one out?

Solution

Because we already pulled out an ace, there are now 3 aces left in the deck. The probability of pulling another ace is $\frac{3}{51}=\frac{1}{17}$.

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General Multiplication Rule

For events $A,B$ which are independent:

$$P(A\cap B)=P(A)\cdot P(B)$$

Joint probability is the probability of both events occurring.

Warning

Be careful! This rule only applies to independent events.

If you are dealing with dependent events, you will need to use the Conditional Probability formula.

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Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred.

We can calculate conditional probability using the following formula:

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$