Business Statistics: Probability, Biases, and Correlation (Week One Study Notes)

Conditional Probability

Definition and Formula

Conditional probability is the likelihood of an event occurring, given that another event has already occurred. It is a fundamental concept in statistics, especially in business decision-making, where outcomes often depend on prior events.

• Conditional Probability Formula:

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

• Key Point: The probability of event A occurring, given that event B has occurred, is equal to the probability of both A and B occurring divided by the probability of B.

• Example: If Andrew Jackson rushes more in a game, he is more likely to score his third run, while Billy Joel who missed the last two is less likely to get the third one.

Common Biases in Probability and Decision-Making

Types of Biases

Human decision-making is often affected by cognitive biases, which can lead to errors in statistical reasoning and business decisions.

• Availability Heuristic: The idea that the events most easily recalled are seen as more likely to happen.

• Conjunction Fallacy: The idea that specific conditions are more probable than general ones.

• Gambler's Fallacy: The mistaken belief that future probabilities are altered by past events, such as believing in a "lucky streak."

• Selection Bias: Drawing generalized conclusions from specific, non-representative examples.

Example: Assuming a coin is "due" to land heads after several tails is an example of the gambler's fallacy.

The Law of Large Numbers

Definition and Application

The law of large numbers states that as a sample size increases, the sample mean will get closer to the population mean. This principle is crucial in business statistics for making reliable predictions and decisions.

• Key Point: Larger samples yield more accurate estimates of population parameters.

• Example: Predicting average customer spending is more accurate with data from 1,000 customers than from 10.

Correlation

Definition and Interpretation

Correlation measures the degree to which two variables are linearly related. It is commonly used in business to identify relationships between factors such as sales and advertising spend.

• Key Point: Correlation does not imply causation. Two variables may move together without one causing the other.

• Example: There may be a negative relationship between elevation and temperature: higher elevation is associated with lower temperature.

Formula for Correlation Coefficient:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

• Additional info: The correlation coefficient $r$ ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no linear relationship.

Summary Table: Common Biases in Probability

Bias Type	Description	Example
Availability Heuristic	Events that are easily recalled are seen as more likely	Believing plane crashes are common after seeing news reports
Conjunction Fallacy	Specific conditions are seen as more probable than general ones	Assuming a person is more likely to be a bank teller and a feminist than just a bank teller
Gambler's Fallacy	Belief that future probabilities are affected by past events	Expecting a coin to land heads after several tails
Selection Bias	Generalizing from non-representative samples	Assuming all customers behave like the most loyal ones