Download Chapter 8A: Probability. A phenomenon or trial is said to be random

Chapter 8A: Probability.
A phenomenon or trial is said to be random if individual outcomes are uncertain. The probability of any outcome
of a random event is the proportion of times the out come would occur in a very long series of repetitions. The
sample space of a random phenomenon is the set of all possible outcomes that cannot be broken down into simpler
components. An event is any outcome or any set of outcomes of a random phenomenon.
A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space
and a way of assigning probabilities to events.
There are a few rules in probability. The ones we are going to focus on are as follows:
1. The probability of an event to happen must be between 0 and 1 (or 0% to 100%). An event with probability
of zero is deemed as impossible, while an event with probability of one is deemed as a certainty.
2. If the probability of an event to happen is P (A), then the probability of that event to not happen is 1 − P (A).
3. If two events A and B are disjoint, then the probability of A or B happening is P (A) + P (B).
4. If two events A and B are not disjoint, then the probability of A or B happening is P (A) + P (B) minus the
probability of both things happening.
5. All possible outcomes together must add up to 1 (or 100%).
Example 1: Coin Tossing. A fair die is a die in which all six sides are equally likely to come show up. Answer the
following:
1.
2.
3.
4.
What
What
What
What
is
is
is
is
the
the
the
the
sample space if you were to toss the die once?
probability that you roll a 7?
probability that you roll an even number?
probability that you roll less than 6?
Example 2: Benford’s Law. Benford’s law, also called the first-digit law, states that in lists of numbers from many
(but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. This peculiar
phenomenon happens very often in naturally occurring data sets, and an explanation of this can be found on p. 256.
By not allowing the first digit to be zero, we have the following probability model:
First Digit
Probability
1
0.301
2
0.176
3
0.125
4
0.097
5
0.079
6
0.067
7
0.058
8
0.051
9
0.046
Answer the following, using the table above:
1.
2.
3.
4.
What
What
What
What
is
is
is
is
the
the
the
the
probability
probability
probability
probability
of
of
of
of
having
having
having
having
a first digit of a one or a two?
a first digit that is not a seven?
an even first digit?
a multiple of three as a first digit?
Section 8B: Combinatorics.
Combinatorics is the study of methods of counting. We start by definition:
The Fundamental Principle of Counting states that if you have a ways of choosing one thing, b ways of choosing a
second after the first is chosen, etc..., then the total number of choice sequences is a · b · . . . .
Example 3: Choosing Toppings. You are at a summer barbecue and you are dressing up your burger. At the
condiment table, you see mustard, mayonnaise, ketchup, and sweet relish all in a squeezable container. Also on the
table, you also see sliced tomatoes, sandwich pickles, and lettuce. How many possible ways can you dress up your
burger?
There are three main combinatorics principle, which will be illustrated in the following examples.
Example 4: DNA Codons. DNA consists of a long sequence of of the nucleotides adenine, cytosine, guanine, and
thymine (abbreviated as A, C, G, T). A codon is a string of three consecutive nucleotides. How many possible codons
can be created with these four nucleotides?
Example 5: Beauty Pageant. Alice, Beth, Christie, Danielle, and Esther are the final five contestants in a beauty
pageant. One will be named second runner up, one will be named first runner up, and one will be named the winner.
How many possible ways can they be arranged do the three placements?
Example 6: State Quarters. You have a four state quarters in your pocket, and the states represented are Tennessee, Rhode Island, Alabama, and Pennsylvania. How many possible combinations are possible if you pick out
three of those four quarters from your pocket?
• The first principle illustrated is an ordered set with replacement. In the example, we can keep reusing our choices
for all the available slots. (AAA is allowed.)
• The second principle illustrated is an ordered set without replacement, which is also called permutations. In this
example, once we use up one choice, we cannot use it again. (AA is not allowed.) The shorthand for permutation is
n Pr , with n being the number of items and r being the number of ordered slots available.
• The third principle illustrated is an unordered set without replacement, which is also called combinations. In this
case, a selection of TAR is the same as a selection of RAT, and ART. The shorthand for combination is n Cr , with
n being the number of items and r being the number in the desired unordered set. For more on these topics, please
see p. 258 - 260.
Your calculator will be of great help here. You should have a combination and a permutation button in your calculator.
However, it will be up to you to determine what the situation calls for. Try the following:
1. A bike lock has four digits, and only one four digit number will unlock the contraption. How many possible
combinations are there?
2. In the Georgia Lottery’s Fantasy Five, five numbers are drawn at random from 39 possible numbers. Matching
these five numbers in any order will mean you won the jackpot. If you cannot make up your mind on your
five numbers, there is a Quick Pick option, in which the computer will randomly choose for you. How many
possible five number sets can quick pick generate?
3. An NBA team has 12 players on its active roster, but only five players can start. There are generally five
positions: point guard (PG), shooting guard (SG), small forward (SF), power forward (PF), and a center (C).
Pretend for now that all 12 players can play at any starting position and all are equally likely to be chosen to
start. How many possible ways can you create a starting lineup?
4. How many possible five digit zip codes do not contain the number zero?
5. A password is considered alphanumeric if it contains both letters (A to Z) and numbers (0 to 9). How many
possible passwords can be created if it must be six characters long and repetition is not allowed?
6. The Miss America pageant has 53 contestants: one from each state, one from Washington, D.C., one from
Puerto Rico, and one from the U.S. Virgin Islands. How many possible ways can ten finalists be selected out
of the 53 contestants?