First of all, congratulations to the following students in our class who have won in the recent school wide Math contest:

Karen Li is the sole 1st place winner in the  6th grade category with $20 prize.

Eugene Jiang, Robert Sheng, Alexander Deng, Jiming Xu are among other prize winners in the 6th grade category and will each get $10 prize.

Congratulations again!

Since we have finished covering the regular content in the textbook, I am adding a little more today. For the last two lessons, we will have review and final exam.

We study combinations and permutations, with the introduction of factorial. A combination is a group of outcomes where the order does not matter, while permutation is a group of outcomes where the order does matter. Factorial is a way of computation. The factorial of a number is the product of the number and all the natural numbers less than the number.

n! = n x (n-1) x (n-3) x … x 3 x 2 x 1

The number of permutations of choosing r things out of n things at a time is denoted as nPr and

nPr = n! / (n-r)!

The number of combinations of choosing r things out of n things at a time is denoted as nCr and

nCr = n! / [r! x (n-r)!]

Today’s material can be found here for Section 10.8.

Homework: download this page

Page 765, #12 – #23.

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.