3) Independent Events, Product Spaces, and the Binomial Density
Independent Events, Product Spaces, and the Binomial Density
Introduction to Independent Events, Product Spaces, and the Binomial Density:
Two events are defined as independent events, if the outcome of one event does not affect the probability of the outcome of the other event.
By using an extension of the conditional probability formulas introduced previously,
P(A|B) = P(A ∩ B) / P(B) and P(B|A) = P(A ∩ B) / P(A)
can be manipulated to show the Multiplication rule for the probability of events A and B,
P(AnB) = P(A|B)*P(B) and P(AnB) = P(A)*P(B|A)
and derive the formula for independent events to become P(AnB) = P(A)*P(B)
For a given process, let S be the set of all possible outcomes. A and B are subsets (events) contained in set S. Therefore, for independent events A and B, the P(AnB) = P(A)*P(B)
The binomial density formula is used to calculate the probability of a certain number of successes in a fixed number of independent trials.
Binomial Distribution Formula P(X = r) = nCr*P(success)^r*P(failure)^(n-r)
Where P(success) + P(failure) = 1
Use Probability Density Bar chart to display corresponding binomial distribution
Questions students should be able to answer:
1) What is the probability of the first event?
2) How to determine if two events are independent?
3) When can the Binomial Distribution Formula be used?
4) How to create a Probability Density Bar chart?