# Section 2: Conditional Probability, Bayes’ Rule, and Independence

### Instructions

• This section covers the concepts listed below.
• For each concept, there is a conceptual video explaining it followed by videos working through examples.
• When you have finished the material below, you can go to the next section or return to the main Mathematical Probability page
Conditional Probability

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. What is the meaning of conditional probability?
2. If $$A$$ and $$B$$ are disjoint and $$P(B)>0$$, what is $$P(A|B)$$? What if they are independent?
3. Is $$P(A|B)=P(B|A)$$? When does it happen?
Law of Total Probability

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. State the law of total probability.
2. Write the law for when $$\Omega=\hspace{.2cm}\displaystyle\cdot \hspace{-10pt}\bigcup_{n=1}^{3}B_n$$.
Bayes’ Formula

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. Why is the Bayes' formula important? How does it help with understanding "cause and effect"?
2. Write the formula when $$\Omega=\hspace{.2cm}\displaystyle\cdot \hspace{-10pt}\bigcup_{n=1}^{3}B_n$$.
Independence of Two Events

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. When are two events independent? What does it mean conceptually?
2. When are two disjoint events independent?
3. Is $$P(A\cap B)=P(A)P(B)$$ always true? Can you provide a counterexample, if not?
Complement of Independent Events

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. If two events are independent, are their complements independent, too?
Independence of $$n$$ Events

### Examples

Directions: The following examples cover the material from the video above.
Bernoulli Trials

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. What is a Bernoulli trial? How many outcomes does it have?
2. How many outcomes are there if we repeat a Bernoulli trial four times? $$n$$ times?
3. We roll a die and record the number that comes up. Is that an example of a Bernoulli trial? What if we only record whether the number is a perfect square or not? If so, find $$p$$ and $$q$$.