Complete our discussion of from Friday - Empirical Rule
Final Remarks on Categorical Data (for now)
- See CategoricalSummaries.R
- Watch the video GoogleDrive/R videos and scripts for Kenyon College/MosaicPlots-final
- One last Example - Simpson's Paradox
Introduction to Probability
Relative Frequency Interpretation
- The probability of an event A, say P(A), is the long-run relative frequency of the event. That is, we independently repeat a random experiment n times and keep track of the fraction of experiments in which the event A occurs. The relative frequency interpretation of P(A) is then the long-run proportion of independent experiments in which A occurs.
- If a random experiment can result in any one of n equally likely outcomes, then the probability of an event A, say P(A), is equal to one over the number of equally likely outcomes. i.e., P(A)=1/n.
- The probability of an event, say P(A), is a number between 0 and 1, inclusive, which measures an individual’s degree of belief in the event.
Using Simulation to Estimate Probabilities
- Demonstration with R (flipping an unbalanced coin 1000 times - see ProbSim.R)
- Small Group Activity - Estimating Probabilities
- Tossing and spinning coins
- Rolling dice
- Dealing cards
Basic Probability Definitions
- The sample space S of a random experiment is the set of all outcomes.
- A random variable is a variable that takes on numerical values determined by the outcome of a random experiment.
- An event is a subset of S.
Example - Rolling a pair of dice
- Identify the sample space.
- Find the probability that the number of dots on the upward facing sides is 6.
Probabilities in a Finite Sample Space
- Assign probabilities to each basic outcome.
- Compute the probability of the event of interest by summing the probabilities of the basic outcomes making up the event.
Addition Rule for Disjoint Events
- Two events, say A and B, are called disjoint events if they have no outcomes in common. If events A and B are disjoint, then P(A or B) = P(A) + P(B).
- Find the probability of getting a 2 or a 6 when rolling two dice.
- If A is any event, the set of basic outcomes that are not in A is called the complement of A and denoted Ac.
- For any event A, the probability that A does not occur is P(Ac) = 1 - P(A)
- Find the probability of getting an even number when rolling a pair of dice.
Multiplication Rule for Independent Events
- Events A and B are independent if knowledge of the occurrence of one event does not change the probability that the other occurs.
- If A and B are independent events, then P(A and B) = P(A) x P(B)
What is the probability of getting two heads when tossing a coin twice?
What is the probability of getting n heads when tossing a coin n times?
Please read sections 3.1 and 3.2 for class on Wednesday
Reminder: Our first quiz will be held on Friday and it will cover the material in Chapters 1 and 2.