We study probabilities for two events that are dependent to each other and we have the following:

If A and B are dependent events, then P(A and B) = P(A) * P(B after A).

We go over quite some examples in the classroom.

Homework:

HomeWorkL13

We continue to study probabilities with more than one events.

Two events, A and B, are said to be mutually exclusive if they cannot occur at the same time. In genenral, if A and B are two mutually exclusive events, then the probability of A or B occurring is: P(A or B) = P(A) + P(B).

Two events are said to be independent events if the occurrence or non-occurrence of one event does not affect the probability of the occurrence of the other event. In general, if A and B are independent events, the probability of both events A and B occurring is the product of their individual probabilities: P(A and B) = P(A) x P(B).

Homework:

Page 66, #9 – #18.

We can use set notation to describe probabilities.

The sample space, usually denoted by S, of a random experiment is the set of all possible outcomes. Thus, each outcome is an element of the sample space,and each event, being a collection of some outcomes, is a subset of the sample space.

Thus P(E) = n(E)/n(S) where n(E) is the number of outcomes in the event E, and n(S) is the number of outcomes in the sample space S.

And we study the basic properties of probabilities: 0 <= P(E) <= 1, and P(E’) = 1 – P(E).

We further study how to use a possibility diagram or a tree diagram to represent a sample space to help us figure out the probability problem.

Homework:

P61, #24 – #29; P65, #3 – #8.

We study the meaning of probability.

Probability is a branch of mathematics that studies the likelihood, or chance, of a phenomenon happening.

A random experiment is a process in which the result cannot be predicated with certainty. The result is called an outcome of the experiment. An experiment is likely to have more than one possible outcome.

A collection of outcomes is called an event. For example, rolling a dice, you get an odd number.

A measure of how likely an event E will take place is called the probability of that event, and it is denoted by P(E). Mathematically, it is defined as:

The probability of an event E, P(E), in a random experiment with equally likely outcomes is:

P(E) = (number of outcomes favorable to event E) / (total number of possible outcomes)

We also distinguish experimental probability from theoretical probability. Experimental probably refers to the probability of an event occurring when an experiment was conducted, as P(E) = (number of times E occurs) / (number of trials).

Theoretical probability, or simply  probability, P(E), is determined by noting all possible outcomes theoretically, and determine how likely the given outcome is for the event E. P(E) = (# of outcomes for event E) / (total # of possible outcomes).

Homework:

P58, #7, 8, 9, 10, 1, 14, 15, 16, 21, 23.

We start the new chapter on Probability of Simple Events. Today we study the notation of set.

We use the term set to mean a collection of well-defined distinct objects. The objects in a set are called elements or members. Note, a set only contains distinct elements, no duplicates, and the order in which the element appear is insignificant.

Two sets A and A are equal, written as A = B, if they have exactly the same elements.

If every element of a set A is also an element of a set B, then A is said to be a subset of B. And if A is a subset of B, but A is not equal to B, then A is said to be a proper subset of B.

We define a universal set as an overall set of a particular problem when all sets for the problem are subsets of this set. Note, different problems may have different universal sets, and a problem can have more than one universal set.

We also define the empty set which contains no elements and is a subset of any set.

Last we define  the compliment of a set A as the set that contains elements that do not belong to the set A but belong to a universal set of A.

Note, pretty everything we study is by definition, so it is important that students understand the definitions.

Homework:

Page 57, #1 – #6, #17 – #20.

We study Mode as another way to represent the center in a data set. The mode of a data set is the value that occurs most often. In a data set, there can be more than one mode, or there can be no mode (when no number occurs more often than others).

We further comparebetween Mean, Median, and Mode.

Homework:

Page 50, #8, #10, #15, #21 – 25.

We study some basic data analysis methods.

There are two common days to measure the center of a data set: the mean, and the median.

The Mean of a data set is the sum of all data values divided by the total number of data.

While the mean tells the average, we need more to know how the data are distributed, or data deviation.

The mean absolute deviation (MAD) of a data set is the total distance of all data values from the mean value divided by the number of values. The bigger this values is, the more scattered the data; the smaller this value is, the more clustered the data.

Mean does not always tell a good story of the average, as some smaller set of data can affect the outcome. Another way to calculate the center is the median, which is the middle value when the data are are arranged in order from the smallest to the largest.

The mean, the median, and the MAD, together they give us better understanding of the data.

Homework:

P50, #4(i)(iii)(v)(vii), #5, #6(b, d, f, h), #7, #16 – #20.