Probability.
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The word probability derives from the Latin probare (to prove, or to test). Informally, probable is one of several words applied to uncertain events or knowledge, being closely related in meaning to likely, risky, hazardous, and doubtful. The theory of probability is the branch of mathematics that studies chances and the longterm patterns of random outcomes.
Have you ever wondered how gambling be business for the casino? It is a remarkable fact that the aggregate result of many thousands of random outcomes can be known with near certainty.
Individual gamblers can never say whether a day at the casino will turn a profit or a loss. But the casino is not gambling. It does not need to load the dice, mark the cards, or alter the roulette wheel. It knows that in the long run each dollar bet will yield it five cents or so of revenue.
The scientific study of probability is a modern development. It began with the study of chances in games and gambling.
The cofounders of probability theory, Pierre de Fermat and Blaise Pascal (1654) were proposed of the “Gambler’s Dispute” problem by a gambler, Chevalier de Méré, who wanted to know whether the payoff in a certain game is fair (more on this later).
Nowadays probability plays the key role in casino business, but it is also heavily used in insurance business, economics, and industry.
In our classroom however, we still use coins, cards, dice, and wheels as examples because their mathematical models are easy to define and their associated experiments can be performed hundreds of times in the classroom.
On the other hand, the mathematical model for life insurance companies is very complicated and we cannot perform experiments in the classroom to collect data for checking the correctness of that model.
If we flipped a normal coin 1000 times and observed that it landed on head 465 times, then we say that
“the relative frequency of getting a head is 465/1000 = 0.465” in these 1000 repetitions of the experiment.
In another terms, we can also say that the “empirical probability” of getting a head is 0.465 (in these 1000 repetitions of the experiment).
The original purpose of building a mathematical theory of probability is to find a formula that can predict the empirical probability of any event to a high accuracy. In addition, we want the prediction to be more accurate when the number of repetitions increases.
If we succeed, we can save many hours from repeating the same experiment millions of times.
Relative frequency = Empirical probability = Experimental probability
These are just different terms for the same thing.
Mathematical models for Probability
For any specific experiment (or random phenomenon) E, it sample space S is the set of all possible outcomes in that experiment.
Example 1. Flipping a coin
In this case, if we do not allow the coin to land on its edge, there will be only two outcomes.
Hence the sample space S = {Head, Tail}
(Note that the size of the sample space is 2, but the sample space itself is not 2.)
Mathematical models for Probability
For any specific experiment (or random phenomenon) E, it sample space S is the set of all possible outcomes in that experiment.
Example 2. Rolling a (6face) die
If we observe just the face landing on top there will be 6 possible outcomes.
Hence the sample space
S = {1, 2, 3, 4, 5, 6}
(Again, note that the size of the sample space is 6, but the sample space itself is not 6.)
Mathematical models for Probability
For any specific experiment (or random phenomenon) E, it sample space S is the set of all possible outcomes in that experiment.
Example 3. Spinning a Roulette wheel
There are 38 slots for the ball to drop into.
Hence the sample space
S = {00, 0, 1, 2, 3, 4, 5, 6, …, 35, 36}
(Again, note that the size of the sample space is 38, but the sample space itself is not 38.)
2
3
10
4
9
8
5
7
6
♠
♠
♠
♠
♠
♠
♠
♠
♠
♠
Q
J
K
Mathematical models for Probability
For any specific experiment (or random phenomenon) E, it sample space S is the set of all possible outcomes in that experiment.
Example 4:
2 cards are drawn simultaneously from the following set.
The sample space will then be
S = {A&K, A&Q, A&J, A&10, A&9, A&8, …,
K&Q, K&J, K&10, …,
… … …
…, 4&3, 4&2, 3&2} and there should be 78 elements in S.
Mathematical models for Probability
A probability model for an experiment E is a mathematical description of E consisting of two parts: a sample space S and a way of assigning probability to its outcomes.
Example 1. Flipping a coin.
If the coin is fair and the person is flipping the coin randomly, then we believe that the head is equal likely to land on top as the tail. Hence
p(Head) = p(Tail) =
Mathematical models for Probability
A probability model for an experiment E is a mathematical description of E consisting of two parts: a sample space S and a way of assigning probability to its outcomes.
Example 2. Rolling a die.
If the die is fair, then each is equal likely to land on top. Hence
p(1 on top) = p(2 on top)= … = p(6 on top) =
Casino Dice are carefully machined, and their drilled holes, called pips, are filled with white material in density equal to the plastic body. This guarantees that the side with 6 pips has the same weight as the opposite side which has only one pip.
Thus each side is equally likely to land upward. All the odds and playoffs of dice games depends on this carefully planned randomness.
Dice balancing Caliper.
Mathematical models for Probability
An event A is any single outcome or a collection of outcomes in the experiment.
In other words, it is a subset of the sample space S.
The probability of an event A, p(A), is the sum of the probabilities of all the outcomes in A.
Example 1:
Let us roll a fair 6face die, and let A be the event of getting an even number on top. Then
p(A) =
In particular, if every outcome in the experiment is equal likely to occur (which is very common assumption), then
p(A) =
Mathematical models for Probability
Example 2:
Let us drop a ball in to a turning roulette, and let A be the event of getting a number between 1 and 18 inclusive.
Since the roulette is almost perfectly balanced, every outcome in the experiment is equal likely to occur
p(ball landing on any specific number) =
Hence
p(between 1 and 18) =
Mathematical models for Probability
Example 3. Spinning the pointer of the following wheel.
The sample space S = {red, yellow, green, blue}
If the pointer is perfectly balanced and the bearing is very smooth, then it is equally likely to stop at any position, hence
p(red) =
p(blue) =
p(green) =
p(yellow) =
(also called the Law of large numbers)
Consider an experiment E in which the theoretical probability of an event A is p. Suppose that the single trial of this experiment is repeated many times, and that the outcome of each trial is independent of the others.
If the number of trials increases, the experimental probability of A will approach the theoretical value p.
Example
Suppose that the theoretical prob of winning a game X is 26%, then
In America during the goldrush era, a very ingenious gambling game garnered a lot of money for its perpetrators.
Three cards were placed in a hat:
one was gold on both sides,
one was silver on both sides, and
one was gold on one side and silver on the other side.
The gambler would take one card and place it on the table showing (for instance) gold on the top side of the card.
Then he would bet the on lookers even money that gold would be on the reverse side, his reasoning being that the card (on the table) could not be sliver/silver, hence there were only two possibilities: gold/silver or gold/gold. A fair and even bet, isn’t it?
The most popular game in state lotteries is Lotto. By 2006, there are only 7 states without Lottos.
To play the California Super Lotto plus you need to pick 5 number from 1 to 47 and one mega number from 1 to 27.
Prior to June 6, 2000, the format was to pick 6 numbers from 1 to 49, hence the nick name 649. This format is still being used in many states.
Lottos are a bad bet, because the state pays out only about half of the money wagered. The only compensation almost all Lotto players receive is the pleasure of dreaming themselves rich.
When a student was walking across the room (with tiles on the floor as illustrated below), a small button fell off from her dress. What is the probability that the (center of the) button landed on a blue tile?
Answer:
Since there are totally 80 square tiles and 20 of them are blue, hence the probability of landing on a blue tile should be
π
5
=
16
π
16
The following target is made up of concentric circle with radii 1, 2, 3, and 4 units. If a dart was thrown randomly and hit the target, what is the probability that it hit the red ring?
Answer:
area of target = π(4)2 = 16 π
area of red ring = π(3)2 – π(2)2
= 5π
4
3
2
Hence
Prob(hitting red ring) =
In some experiments it is inefficient to list all the outcomes in the sample space. Therefore, we develop alternative procedures to compute probabilities such as drawing a tree diagram.
A Tree diagram is a diagram consisting of line segments connected like the branches and twigs of a tree. In particular, there is never a loop in a tree diagram.
The starting point of a tree diagram is called the root. Each branching point is called a node.The number of levels in a tree diagram is equal to the number of steps in the corresponding experiment.
H
T
H
H
T
T
H
H
T
T
H
T
T
3rd time
2nd time
1st time
Tree Diagram
for the experiment
“a coin is flipped 3 times”
start
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
H
H
T
T
H
H
H
H
T
T
T
T
H
H
H
H
T
T
T
T
H
H
T
T
T
T
A coin is flipped 5 times
H
start
T
The orange path represents the sequence of THTHH
Probability trees and one stage experiments
If we label each branch of the tree with the appropriate probability, then we get a probability tree.
red
A ball is drawn from the following jar at random.
2/9
3/9
green
start
4/9
blue
2
2
4
3
1
2
2
4
3
3
9
9
8
8
8
8
8
8
8
8
8
3
9
Blue
Blue
Blue
Red
Red
Red
Green
Green
Green
Complex Experiments and Probability TreesStart
A Jar contains: 2 red balls, 3 green balls, and 4 blue balls.
If two balls are taken out sequentially and randomly without replacement, what is the probability of getting two balls of the same color?
Blue
Red
Green
Reset
Reset
Reset
The Los Angeles Lakers and Portland Trailblazers are going to play a “best 2 out of 3” series. Suppose that the probability that the Lakers win an individual game with Portland is 3/5, draw a probability tree to show possible outcomes.
LA
3/5
LA
3/5
LA
3/5
2/5
P
P
2/5
3/5
LA
LA
3/5
2/5
P
P
2/5
2/5
P
Suppose that an experiment consists of a sequence of simpler experiments. Then the probability of each final outcome is equal to the product of the probabilities of the simpler experiments that make up the sequence.
Example:
Suppose that we roll a regular die twice. Then
Prob(rolling a 3 followed by rolling a 5) = Prob(rolling a 3) × Prob(rolling a 5) = =
Suppose that an event A is the union of two (or more) mutually exclusive simpler events A1, A2. Then Prob(A) = Prob(A1) + Prob(A2)
Example:
Suppose that we roll 2 dice simultaneously. Then Prob(getting a sum of 11) = Prob(rolling a 5 on the 1st die and rolling a 6 on the 2nd die) +
Prob(rolling a 6 on the 1st die and rolling a 5 on the 2nd die)
=
=
Multiplication rule
If events A and B are independent, then the probability that both events occur (either simultaneously or one after the other) is
P(A and B) = P(A)×P(B)
"A gambler's dispute in 1654 led to the creation of a mathematical theory of probability by two famous French mathematicians, Blaise Pascal and Pierre de Fermat. Antoine Gombaud, Chevalier de Méré, a French nobleman with an interest in gaming and gambling questions, called Pascal's attention to an apparent contradiction concerning a popular dice game. The game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money (i.e. 1 to 1 payoff) on the occurrence of at least one "double six" during the 24 throws.
A seemingly wellestablished gambling rule led de Méré to believe that betting on a double six in 24 throws would be profitable, but his own calculations indicated just the opposite.
Permutations
An ordered arrangement of objects is called a permutation.
For example, the permutations of the letters C,A,S,T are
ACST CAST SACT TACS
ACTS CATS SATC TASC
ASCT CSAT SCAT TCAS
ASTC CSTA SCTA TCSA
ATCS CTAS STAC TSAC
ATSC CTSA STCA TSCA
You can see that there are 4×3×2×1 = 24 many permuations.
Fundamental Counting Property
If an event A can occur in r ways, and for each of these r ways, an event B can occur in s ways*, then event A and B can occur, in succession, in r×s ways.
* this condition can also be rephrased as “the number of outcomes in event B is independent of event A.”
Example
Suppose that in a local diner, a supper consists of a starter, an entrée, and a beverage.
If there are 3 choices for the starter, 5 choices for the entrée, and 7 choices for beverages, how many different suppers can be created?
1. The license plates in Utah consist of 3 digits followed by 3 letters. How many such license plates are possible?
Passenger vehicle License Plates in California
If the 1st and the 3rd letters cannot be an I and O, how many possible combinations are there?
2. Given the set of digits {5, 6, 7, 8, 9}, how many 4digit numbers can be formed such that a) the digits are different? b) the digits are different and the number is divisible by 5? c) the digits are different and the number is > 6000?
d) the digits are different and the number is < 8000?
The number of permutations for n different objects is 1×2×3×4× ··· × n
The factorial notation
The product 1×2×3×4× ··· × n is called nfactorial and is written as n!.
Examples
1! = 1
2! = 1×2 = 2
3! = 1×2×3 = 6
……
10! = 3,628,800
Remark: 0! is defined to be 1.
1. Miss Murphy wants to seat her 12 students in a row for a class photo. How many different seating arrangements are there?
Answer: 12!
2. Seven of Miss Murphy’s students are girls and 5 are boys. In how many different ways can she seat the 7 girls on the left, then the 5 boys on the right?
Answer: 7! × 5!
Permutations of a set of objects taken from a larger set
Example:
In a certain lottery game, four different digits are taken from the digits 1 to 9 to form a 4digit number. How many different numbers can be made?
Answer: 9×8×7×6
This answer can also be written as 9!/5!
Theorem
The number of permutations of r objects taken from n (≥ r) objects is
nPr =
A collection of objects, in no particular order, is called a combination.
Example
Suppose that there are 5 ingredients – Pepperoni, sausage, green pepper, olive, and mushroom – three are chosen to make a pizza. How many possible combinations are there?
Pepperoni, Sausage, Green pepper, Olive, Mushroom.
PSG,
PSO,
PSM,
PGS,
PGO,
PGM,
POS,
POG,
POM,
PMS,
PMG,
PMO,
SPG,
SPO,
SPM,
SGO,
SGM,
SGP,
SOP,
SOG,
SOM,
SMP,
SMG,
SMO,
GPS,
GPO,
GPM,
GSO,
GSP,
GSM,
GOP,
GOS,
GOM,
GMP,
GMS,
GMO,
OPS,
OPG,
OPM,
OSP,OSG,
OSM,OGP,OGS,OGM,OMP,
OMS,
OMG,
MPS,
MPG,
MPO,MSP,MSG,
MSO,MGP,MGS,MGO,MOP,MOS,MOG,
We can see that every combination repeats 6 times. Hence we need to divide the answer by 6.
!
6
!
=
=
15

´
´
(
6
2
)!
2
!
4
!
2
!
20
!
20
!
=
=
125
,
970

´
´
(
20
12
)!
12
!
8
!
12
!
Theorem
The number of combinations of r objects chosen from n objects, where 0 ≤ r ≤n, is
[Note: Occasionally, nCr is denoted and read “n choose r”]
0C0
1C01C1
2C02C12C2
3C03C13C23C3
4C04C14C24C34C4
. . . . .
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1. A coin is tossed 7 times, how many ways are there to get 3 heads and 4 tails?
2. A candy store has 24 different kinds of candies. How many ways can you choose 3 different types?
3. In a class of 16 girls and 14 boys, how many ways are there to form a committee of 5 girls and 4 boys?
4. From a standard deck of cards, how many ways can we make one pair?
Answer: (4C2)×13
5. From a standard deck of cards, how many ways can we make a flush poker hand? (a flush means all 5 cards in the same suit.)
Answer: (13C5)×4
Example 1.
In Mrs. O’Neill’s class, there are 20 students. If 16 are girls, how many are boys?
Example 2.
In a year of 365 days, if 104 days are weekends and holidays that you don’t work, how many day do you have to work?
Each student ID is a 5digit number (including 00000), how many of these have duplicate digits?
Finding probability by subtraction.
Example 1.
In a certain raffle, the chance of winning is only 0.01%, what is your chance of losing?
Example 2.
The probability of raining tomorrow is 25%. What is the probability that there is no rain tomorrow?
If 4 cards are drawn from a standard deck randomly, what is the probability that they are from different suits?
Example 4
If 4 cards are drawn from a standard deck randomly, what is the probability that some are from the same suit?
Permutations of not totally distinguishable objects
Suppose that you have3 indistinguishable yellow tulips,5 indistinguishable red tulips,
1 purple tulip,
2 indistinguishable pink tulips.How many different ways can you arrange them in a row?
!
´
´
´
3
!
5
!
1
!
2
!
Permutations of not totally distinguishable objects
Suppose that you have3 indistinguishable yellow tulips,5 indistinguishable red tulips,
1 purple tulip,
2 indistinguishable pink tulips.How many different ways can you arrange them in a row?
Answer:
Since there are totally 11 tulips we have
Another example of the same type
How many ways can we rearrange the letters in the word “MISSISSIPPI” to get a different string of 11 letters?
Answer: Since there are 11 letters totally, 1M, 4S’s, 4 I’s, and 2 P’s, there are
11
!
´
´
´
1
!
4
!
4
!
2
!
different arrangements.
48
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»
$
10
$
0
.
50
$
0
.
308
52
52
Mathematical ExpectationsThe expected value of an experiment E is the average amount one expects to get when the experiment is repeated a large number of times.
Example
Suppose that in a certain game, you may draw a card from a standard deck. You will be paid $10 if you get an ace, otherwise you have to pay 50 cents.If you were to play this game a large number of times, what will be your average winnings per game? Should you play this game often for some extra income?
Answer: expected value =
and yes, you can play this game more often to get extra income.
The expected value is not necessarily a possible outcome in the experiment.
For example, the expected value of rolling a normal sixsided die is
and we know that it is impossible to roll a 3.5 from a normal die.
This is similar to the statistics that the “average number of children in an American family is 2.3”.
Mathematical expectation is an important application of probability in gambling, industry, insurance business, and many other practical fields.
We are going to see several examples right afterwards.
1
×

×
=
$
1
$
35
$
0
.
05
38
38
Example 1
A U.S. roulette wheel has 38 slots. If you bet on any single number such as “8”, the casino will pay you 35 to 1. In other words, if you bet $1 on the number “8”, and you win, the casino will return your $1 and pay you $35. If you lose, you lose $1.
What is the expected value of this game in the eyes of the casino?
Answer:
Expected value =
This means on average, the casino can get $0.05 for each dollar bet on wheel.
Suppose that an insurance company has broken down yearly automobile claims for drivers from age 16 through 21, as shown in the table below. How much should the company charge as its average premium in order to break even on its cost for claims?
Answer:
Exp = $0×(0.80) + $2000×(0.10) + $4000×(0.05) + $6000×(0.03) + $8000×(0.01) + $10,000×(0.01)
= $760
Walt, who is a realtor, knows that if he takes a listing to sell a house, it will cost him $1,000. However, if he sells the house, he will receive 6% of the selling price. If another realtor helps him to sell the house, he will get 3% of the selling price. If the house remains unsold after 3 months, he will lose the listing and receive nothing.
Suppose that the probability for selling $200,000 house are as follows:
prob(Walt sells the house) = 0.4
prob(another agent sells the house) = 0.2
prob(house unsold after 3 months) = 0.4
What is Walt’s expectation if he takes the listing?
Solution:
6% of $200,000 = $12,000
3% of $200,000 = $6,000
Expected value = $12,000×0.4 + $6,000×0.2 – $1,000
= $5,000