Intro to Business Statistics: Probability - Rules of Probability, Complement Rule Addition Rule for Mutually Exclusive Events Addition Rule for NonMutually Exclusive Events

What This Is

The Rules of Probability are essential in business decisions, as they help us understand the likelihood of events occurring. A retail chain wants to know if average daily sales exceed $10,000 to determine if they should increase inventory. By applying the Rules of Probability, they can make informed decisions about inventory levels, pricing, and marketing strategies.

Key Formulas & Symbols

• P(A-B) = P(A) + P(B) where P(A) = probability of event A, P(B) = probability of event B.
• P(A-B) = P(A) × P(B) where P(A-B) = probability of both events A and B occurring.
• P(A') = 1 - P(A) where P(A') = probability of the complement of event A.
• P(A-B) = P(A) + P(B) - P(A-B) where P(A-B) = probability of either event A or event B occurring.
• P(A-B) = P(A) + P(B) - P(A-B) where P(A-B) = probability of both events A and B occurring.
• P(A-B') = P(A) - P(A-B) where P(A-B') = probability of event A occurring but not event B.
• P(A'-B) = P(B) - P(A-B) where P(A'-B) = probability of event B occurring but not event A.

Step-by-Step Procedure

1. Identify the events: Clearly define the events A and B.
2. Determine if events are mutually exclusive: If events cannot occur at the same time, they are mutually exclusive.
3. Apply the appropriate rule: Use the Complement Rule, Addition Rule for Mutually Exclusive Events, or Addition Rule for Non-Mutually Exclusive Events.
4. Compute the probability: Plug in the values and calculate the probability.
5. Interpret the result: Understand the meaning of the calculated probability in the context of the business decision.

Common Mistakes

• Mistake: Using the Addition Rule for Mutually Exclusive Events when events are not mutually exclusive.
• Correction: Use the Addition Rule for Non-Mutually Exclusive Events, which takes into account the probability of both events occurring.
• Mistake: Misinterpreting the Complement Rule as the probability of an event occurring.
• Correction: The Complement Rule gives the probability of the event not occurring, not the probability of the event occurring.
• Mistake: Not considering the degrees of freedom when applying the Addition Rule for Non-Mutually Exclusive Events.
• Correction: Make sure to account for the degrees of freedom when calculating the probability.

Quick Practice Problems

1. A retail chain wants to know if average daily sales exceed $10,000. If the probability of sales exceeding $10,000 is 0.2, what is the probability that sales do not exceed $10,000? Answer: 0.8, because P(A') = 1 - P(A).
2. A company wants to know if the probability of a customer purchasing a product is 0.3. If the probability of a customer not purchasing the product is 0.2, what is the probability that a customer purchases the product and also has a warranty? Answer: 0.06, because P(A-B) = P(A) × P(B).
3. A marketing firm wants to know if the probability of a customer responding to an ad is 0.4. If the probability of a customer not responding to the ad is 0.3, what is the probability that a customer responds to the ad or has a positive attitude towards the brand? Answer: 0.73, because P(A-B) = P(A) + P(B) - P(A-B).

Last-Minute Cram Sheet

• P(A-B) = P(A) + P(B) for mutually exclusive events.
• P(A-B) = P(A) + P(B) - P(A-B) for non-mutually exclusive events.
• P(A') = 1 - P(A) for the complement rule.
• P(A-B) = P(A) × P(B) for independent events.
• ? = 0.05 is the default significance level.
• p-value is NOT the probability that H? is true – it’s the probability of observing the data (or more extreme) if H? is true .
• Degrees of freedom matter when applying the Addition Rule for Non-Mutually Exclusive Events .
• Mutually exclusive events cannot occur at the same time.
• Non-mutually exclusive events can occur at the same time.
• Independent events have no effect on each other.