Week 9, Day 1

Today we learned about probability in its various forms.

Independent events are when the occurrence of one event does not affect the occurrence of the 2nd event. The complement of an event is the probability an event does not occur. A happening whose result is uncertain is called an experiment. The possible results of the experiment are outcomes, and the set of all possible outcomes of the experiment is the sample space. Any subcollection of a sample space is an event. The measure of the likelihood that an event will occur based on chance is called the probability of an event. If an event (E) has n(E) equally likely outcomes and its sample space (S) has n(S) equally likely outcomes, the probability of event (E) is P(E) = n(E) / n(S). The probability of an event must be between 0 and 1. If P(E) = 0, the event cannot occur, and (E) is called an impossible event. If P(E) = 1, the event (E) must occur, and (E) is called a certain event. 

Example 1:

A box contains 3 red marbles, 5 balck marbles, and 2 yellow marbles. If a marble is selected at random from the box, what is the probability that it is yellow?

n(E) / n(S) = 2 / 10 = 1 / 5 = 20%

Mutually Exclusive Events:

The probability of an event must be between 0 and 1. If P(E) = 0, the event E cannot occur, and E is called an impossible event. If P(E) = 1, the event E must occur, and E is called a certain event. Two events A and B, from the same sample space, are mutually exclusive if A and B have no outcomes in common. If A and B are events in the same sample space, the probability of A or B occurring is given by P(A U B) = P(A) + P(B) - P(A inverted U B). U refers to union, and inverted U refers to intersection.

Example 2: 

A box contains 3 red marbles, 5 black marbles, and 2 yellow marbles. If a marble is selected at random from the box, what is the probability that it is either red or black.

3 / 10 + 5 / 10 = 8 / 10 = 4 / 5 = 80%

Independent Events: 

If A and B are independent are independent, the probability that both A and B will occur is P(A and B) = P(A) + P(B). That is, to find the probability that two independent events will occur, multiply the probabilities. Let A be an event and let A' be its complement. If the probability of A is P(A), the probability of the complement is P(A') = 1 - P(A). "With relplacement" implies independent events.

Example 3:

A box contains 3 red marbles, 5 black marbles, and 2 yellow marbles. If two marbles are randomly selected with replacement, what is the probability that both marbles are yellow? 

2 / 10 x 2 / 10 = 4 / 100 = 2 / 50 = 4%