Question

What is probability, what can it values be, and what those values can mean (i.e. unusual, rare, likely, impossible, certain, uncertain). Also, what are the differences between empirical probability and theoretical probability. Next what is the addition rule and the multiplication rule. How can you tell the difference between them and when to use them? Also, what do the terms mutually exclusive/disjoint and independence and dependence events mean? How do they differ?

Answer 1

Probability is the measure of the likelihood that an event will occur in a Random Experiment. Probability is quantified as a number between 0 and 1. The higher the probability of an event, the more likely it is that the event will occur. An event with a probability of 1 can be considered a certainty. An event with a probability of .5 can be considered to have equal odds of occurring or not occurring. An event with a probability of 0 can be considered an impossibility.

The theoretical probability of an event occurring is an "expected" probability based upon knowledge of the situation. It is the number of favorable outcomes to the number of possible outcomes. The empirical (or experimental) probability of an event is an "estimate" that an event will occur based upon how often the event occurred after collecting data from an experiment in a large number of trials. When doing an experiment, the theoretical probability refers to what we think might happen before starting the experiment. Empirical probability refers to what actually happens in the experiment. Empirical probability is what we get after we conduct an experiment with a large number of trials. It is a way of predicting whether or not an event will happen based on data collected when performing an experiment having lots of trials. Theoretical probability is a simpler concept. It does not involve any actual experiment. It is mostly used when we know the sample space and when the events in the sample space are all equally likely.

Additional Rule: When two events, A and B, are non-mutually exclusive, the probability that A or B will occur is:

P(A or B) = P(A) + P(B) - P(A and B).

Multiplication Rule: The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred.

P(A ∩ B) = P(A) P(B|A)

Tell the difference between the two. Provide the notations and then tell me what type of problem I would use each one for.

Addition Rule: Notation for Addition Rule: P(A or B) = P(event A occurs or event B occurs or they both occur).

Multiplication Rule: Notation: P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial).

Mutually exclusive or disjoint event: Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint. If two events are disjoint, then the probability of them both occurring at the same time is 0. Disjoint: P(A and B) = 0.

Independence event: The occurrence of one event does not affect the probability of the occurrence of another.

Dependence event: If the occurrence of one event does affect the probability of the other occurring, then the events are dependent.

According to definition, mutually exclusive events are those which have no outcomes in common. Independent events are those, between whom the occurrence of one does not affect the occurrence of the other. Dependent events are those, between whom the occurrence of one does affect the occurrence of the other.