Basic probability: Joint, marginal and conditional probability | Independence

Summary

In this video, the concept of basic probability is explored through the analysis of categorical variables, such as gender and favorite HBO shows. The video explains joint, marginal, and conditional probabilities, using a survey of 500 subscribers to HBO. It demonstrates how to calculate the probability of various outcomes, including the chance of a subscriber being male or preferring a particular show. The video also covers the concept of independence between two categorical variables, showing how to assess whether gender and show preference are independent. The lesson helps viewers understand foundational concepts in probability with a practical example.

Takeaways

• 😀 **Joint Events**: When two variables interact, such as gender and favorite shows, their combined outcomes are called joint events and are represented in a table.
• 😀 **Probability Distribution**: A joint probability distribution is created by dividing the counts from the survey data by the total number of observations (500 in this case), resulting in probabilities instead of raw counts.
• 😀 **Marginal Probability**: This is the probability of one event occurring (e.g., the probability of being male or liking *Game of Thrones*), calculated by summing joint probabilities over one variable.
• 😀 **Conditional Probability**: This refers to the probability of an event occurring given that another event has already happened, such as the probability of liking *Game of Thrones* if the person is female.
• 😀 **Calculating Conditional Probabilities**: To find a conditional probability, you focus on the subset of data that matches the given condition, such as female respondents, and divide the joint probability by the probability of the condition.
• 😀 **Independence Test**: Two variables are considered independent if the joint probability of both events equals the product of their marginal probabilities. If not, the variables are dependent.
• 😀 **Union of Events**: The probability of either of two events occurring is found by summing their individual probabilities and subtracting the joint probability to avoid double-counting.
• 😀 **Probability of Male or Westworld**: The probability of being male or preferring *Westworld* requires summing the joint probabilities where either condition is met.
• 😀 **The Importance of Marginal Probability Distributions**: These distributions give insights into the individual likelihood of each event occurring, independent of other factors.
• 😀 **Not Independent Variables**: In this example, gender and favorite show are not independent, as the conditional probabilities (like the likelihood of a female liking *Game of Thrones*) differ significantly from the marginal probabilities.

Q & A

What is the difference between joint probability and marginal probability?

Joint probability refers to the likelihood of two events occurring together, such as the probability of a person being male and preferring Game of Thrones. Marginal probability, on the other hand, refers to the probability of a single event occurring, ignoring other variables, such as the probability of a person being male or the probability of preferring Game of Thrones.

What does the term 'marginal probability distribution' refer to?

The marginal probability distribution is the probability distribution of a single variable, ignoring the other variables in the data. In the video, it refers to the distribution of show preferences or gender, considering only one variable at a time.

What is a conditional probability, and how is it calculated in the video example?

Conditional probability is the probability of an event occurring given that another event has already occurred. In the video, the conditional probability is calculated by dividing the joint probability of two events by the probability of the condition. For example, the probability of someone liking Game of Thrones given they are female is calculated as the joint probability of being female and preferring Game of Thrones divided by the probability of being female.

How do you interpret the intersection symbol (∩) in probability?

The intersection symbol (∩) represents the occurrence of two events happening together. In the video, it is used to indicate joint probability, such as the probability of a person being both female and preferring Game of Thrones.

Why do joint probability distributions sum to 1?

Joint probability distributions sum to 1 because they account for every possible outcome in the sample space. In the video, all 500 subscribers are represented in the distribution, and since everyone falls into one of the possible combinations of gender and show preference, the total probability must equal 1.

What does the union symbol (∪) represent in probability?

The union symbol (∪) represents the occurrence of either one event or another, or both. In the video, it is used to calculate the probability of a person being male or preferring Westworld. It includes all joint probabilities where a person is either male or likes Westworld.

What is the formula for calculating the union of two events in probability?

The formula for calculating the union of two events A and B is: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). This formula is used to avoid counting the intersection twice, as shown in the video when calculating the probability of being male or preferring Westworld.

How do you assess whether two variables are independent based on conditional probability?

Two variables are considered independent if the conditional probability of one variable given the other is equal to the marginal probability of the first variable. In the video, it was shown that the probability of preferring Westworld given gender is different from the marginal probability of preferring Westworld, indicating that gender and show preference are not independent.

What is the main takeaway about the relationship between gender and HBO show preferences from the video?

The video demonstrates that gender does affect show preferences on HBO. The conditional probability distributions for males and females show differences in show preferences, indicating that the two variables (gender and show preference) are not independent.

Why do the probabilities of liking a show differ between the general population and females in the video?

The probabilities differ because when gender is considered as a condition, the preferences shift. In the video, females are more likely to prefer Game of Thrones and less likely to prefer Westworld compared to the general population. This suggests that gender influences show preferences.