EXAMPLES:
DETERMINING OUTCOMES
1.
For the experiment of tossing a coin, what are the possible
outcomes?
2.
For the experiment of tossing two coins, what are the possible
outcomes?
3.
For the experiment of rolling a die, what are the possible outcomes?
4.
For the experiment of rolling two dice, what are the possible
outcomes?
PROBABILITY DEFINITIONS
OUTCOME
Outcomes are the possible results of
an experiment or observation.
EVENT
An event consists of one or more
outcomes that share a property of
interest.
PROBABILITY
The probability of an event, written,
P(event), is a number that tells us how
likely it is that the event will occur.
An event that is impossible has a
probability of 0.
An event that is SURE to happen has a
probability of 1.
All other events have a probability
between 0 and 1.
FINDING THEORETICAL PROBABILITIES
STEP 1:
Count the total number of possible outcomes.
STEP 2:
Count the number or “favorable” outcomes– that is, the
number of outcomes that correspond to the event of
interest.
STEP 3:
Determine the probability of the event by:
P(event) = # of “favorable” outcomes
total # of outcomes
EXAMPLES:
1.
What is P(one head comes up) in two tosses of a fair coin?
2.
What is P(a 2 and a 6 come up) when two dice are tossed?
3.
When a single card is randomly drawn from a standard deck of 52
cards, what is P(a club is drawn)?
4.
If I draw a marble fro a bag containing 3 red marbles, 2 blue
marbles, and 1 yellow marble, what is P(a blue is drawn)?
5.
What is P(the sum of the dice is 4) when two dice are tossed?
6.
What sum has the highest probability when two dice are rolled?
7.
In a family with three children, what is the probability that, all
three children are girls? (Assume that having a girl and having a
boy are equally likely.)
What is the P(two girls) in a family of three children?
THEORETICAL PROBABILITY:
The theoretical probability of a chance event tells how likely
that event is– IN THEORY. Theoretical probabilities tell us
what to expect in an experiment when the experiment is up
to chance.
EMPIRICAL PROBABILITY:
Empirical probabilities are based on statistical
observations or experiments. The empirical probability of
an event is the relative frequency of that event (as a
decimal.)
SUBJECTIVE PROBABILITY:
Subjective probability is an estimated probability, based on
experience or intuition.
EMPIRICAL PROBABILITY EXAMPLES
1.
A certain river has flooded 6 times in the last 1000 years.
What is the probability of a flood in any given year?
2.
The following histogram shows the times between 300
eruptions of the Old Faithful geyser in Yellowstone
National Park.
50
60
70
80
90
100
110
If you walk up right after Old Faithful erupts, what is the
probability that you will have to wait less than an hour to
see the next eruption?
3. If I think that a certain die is weighted to favor rolling a 6,
how can empirical probabilities be used to test whether the
die is, in fact, weighted?
If I roll the die 10,000 times and get 2,872 sixes, do I
have reason to suspect the die is weighted?
ODDS:
The odds of an event A are given by:
# of outcomes corresponding
to the event A
# of outcomes corresponding
to the event (not A)
EXAMPLES:
1. Find the odds of rolling a 6 in a single toss of a die.
2. Find the odds of getting heads when a coin is flipped.
3. Find the odds of rolling doubles when two dice are rolled.
4. Find the odds of getting one head and one tail when two coins are
flipped.
5. If the probability of a certain event is 2/5, what are the odds of that
event?
RULE FOR OR
P(A or B) = P(A) + P(B) – P(A and B)
If A and B are non-overlapping events, then
P(A or B) = P(A) + P(B)
RULE FOR AND
P(A and B) = P(A) H P(B given A)
If A and B are independant events, then
P(A and B) = P(A) H P(B)
“OR” EXAMPLES:
1. If a card is drawn from our set of overhead cards
(red, green or blue numbered 1 - 5), find the
probability that the card drawn is red or blue.
2. If a card is drawn from a standard deck of 52
cards, find the probability that the card is an ace or
a diamond.
3. If two dice are rolled, find the probability that the
sum is 6 or doubles are rolled.
4. If two dice are rolled, find the probability that the
sum is 3 or the sum is 9.
“AND” EXAMPLES:
1. What is the probability of drawing 2 Snickers bars
in a row from a bag containing 6 Milky Ways and 3
Snickers?
2. What is the probability of drawing 3 Snickers bars
in a row from the same bag?
3. What is the probability of rolling three 6's in three
rolls of a die?
“OR” AND “AND” PROBABILITY EXAMPLES:
1.
Find the probability that 3 randomly selected people all have telephone numbers
ending in 0.
2.
Find the probability of drawing an ace or a heart from a shuffled standard deck of
52 cards.
3.
Find the probability of drawing two aces in two successive draws from a shuffled
standard deck of 52 cards.
4.
A certain wrench has a 4% breakage rate in normal use. Another brand of wrench
has a 2% breakage rate. What is the probability that a person who uses both
wrenches moderately often will break both during normal use?
5.
When two dice are tossed, find the probability that either doubles are rolled or the
sum is 10.
6.
In a room with 4 Democrats and 6 Republicans, what is the probability of selecting
2 Republicans when 2 people are randomly selected?
7.
Find the probability of rolling 4 successive sixes with four rolls of a fair die.
8.
What is the probability of selecting either a woman or a Canadian when you select a
person randomly from a group composed of 10 Canadian women, 5 Canadian men, 8
American women, and 9 American men?
9.
When a single die is tossed, what is the probability of rolling an even number or a 3?
10.
Find the probability of drawing an diamond or a heart from a shuffled standard deck
of 52 cards.
11.
Three soccer teams will finish first, second and third in a field of 12 evenly matched
teams, which includes 5 European teams. What are the chances of a European
medal sweep?
12.
In POWERBALL, the first 5 numbers are randomly selected between 1 and 49. There
are 1,906, 844 such combinations of five numbers. The powerball is randomly
selected between 1 and 42. What is the probability of winning the game? (That is,
what is the probability of selecting the correct 5 number combination and the
correct power ball number.)
“AT LEAST ONE” RULE
If all trials are independent, then
P(at least one event A in n trials) = 1 – [P(not A in one trial)]n
“AT LEAST ONE” RULE EXAMPLES:
1. Find the probability that you will get at least one
head in five tosses of a fair coin.
2. What is the probability that at least one 6 will come
up in 4 rolls of a fair die?
3. In a certain city, there are 2 suicides a week, on
average. What is the probability of at least one
suicide in the next 10 days?
4. Find the probability that you will get at least one
double six in 24 rolls of a pair of fair dice.
5. In 7 tosses of a pair of dice, what is the probability
that the numbers on the dice will sum up to 10 for at
least one of the tosses?
BIRTHDAY PROBLEMS:
1.
In a room of 30 people, what is the
probability that there is at least one
other person in the room who has the
same birthday as you?
2.
In a room of 30 people, what
is the probability that there
are at least two people in the
room who share the same
birthday?
The Law Of Averages
If an experiment is repeated many times, the proportion
of times that the event A occurs will approach P(A).
(The more times the experiment is repeated,
the closer the proportion should be to P(A).)
The GAMBLERS FALLACLY
The GAMBLERS FALLACLY is the mistaken belief that a streak
of bad luck makes a person due for a streak of good luck.
GAMBLER’S FALLACY EXAMPLE:
A roulette wheel has 18 black numbers, 18 red
numbers, and the numbers 0 and 00 in green.
Suppose that you bet on red, and lose four of the
first five times you play. Should you expect to
recover your losses if you keep playing? Why or why
not?
EXPECTED VALUE
The expected value of an experiment is the
sum of the values of the different outcomes of the
experiment multiplied by their corresponding
probabilities.
1. Suppose someone offers you 10 to 1 odds that
you will not get three heads when you toss a
coin 3 times. That is, you lose $1 if you fail, and
you win $10 if you succeed. Is that a good bet
for you? (What is the expected value of the bet).
2. Usually in roulette, betting on a red is a 1 to 1
bet. That is, if you win, you gain the amount you
bet; if you lose, you lose the money you bet.
What is the expected value of a $10 bet on red?
3. An insurance policy sells for $500 per year.
Based on past data, an average of 1 in 50 policy
holders will file a $5000 claim, 1 in 100 a
$10,000 claim, and 1 in 200 a $30,000 claim in a
given year. Find the expected value to the
company per policy sold.
HOUSE EDGE
Casinos make money because the expected value in
every game is negative (for the players.)
The house edge is the amount the casino can expect to
earn per dollar that is bet.
1.
Averaged over all games and all bets being played, a
particular casino has a house edge of 0.08 (8 cents
per dollar gambled). If a total of $150 million is bet in
the casino over the course of a year, what is the
casino’s total profit?
2.
A casino card game takes $4 bets. Players have a
30% chance of winning (in which case they win back
their bet plus $8), and they have a 70 % chance of
losing their bets. What is the house edge?
3.
In a certain casino, the house wins on blackjack tables
with a probability of .507. All bets at blackjack are 1 to
1. That is, if you win, you gain the amount you bet; if
you lose, you lose the money you bet. What is the
house edge in blackjack?
4.
A roulette wheel has 18 black numbers, 18 red
numbers, and the numbers 0 and 00 in green. Usually
in roulette, betting on a red is a 1 to 1 bet. What is the
house edge in roulette?
ARRANGEMENTS WITH REPETITION
EXA:
Suppose that your ATM password is a 4-digit number
(chosen from the digits 0-9). How many such passwords
are possible?
RULE FOR COUNTING
ARRANGEMENTS WITH REPETITION
If we select r things from n choices
(and we can select the same thing more than once),
r
then n different arrangements are possible.
1.
How many 4-character passwords are possible if the
characters are chosen from the letters A-Z and the
numbers 0-9?
2.
How many different social security numbers are possible?
(Social security numbers are 9 digits long.)
3.
Suppose a certain password must be two digits followed by
four letters. How many such passwords are possible?
4.
In Utah, license plates have 3 single-digit numbers followed
by three letters. How many such license plates are
possible?
PERMUTATIONS
EXA:
In how many ways could we pick a president, vice-president,
and secretary from a group of 5 people?
RULE FOR
PERMUTATIONS
If we select r things a group of n things, where no item
may be selected more than once and we care about the
different orderings, the total number of possible orderings
(permutations) is:
PERMUTATION EXAMPLES:
1.
Using the digits 1, 3, 5, 7, and 9, with no repetition of digits,
how many different 3-digit numbers can be made?
2.
Suppose that a certain combination lock has 45 numbers and
that the combination must consist of three different numbers.
How many different combinations are possible on this lock?
3.
In a golf tournament of 12 golfers, how many different possible
orderings of the golfers are there?
COMBINATIONS
EXA:
In how many ways can I pick two numbers from the numbers
1, 2, 3, and 4 if the order of the choices doesn’t matter
(i.e. the choice 1, 2 is the same as the choice 2,1)?
RULE FOR
COMBINATIONS
If we select r things a group of n things, where no item
may be selected more than once and the order of the
arrangement doesn’t matter, the number of possible
combinations is:
COMBINATION EXAMPLES:
1.
In how many different ways could a 3-person committee be
chosen from a group of 10 people?
From a group of 4 people?
2.
How many different 5-card hands can be dealt from a
standard deck of 52 cards?
3.
Suppose that in POWERBALL, the first five balls are chosen
from a set of balls labeled with the numbers 1-49. How
many possible combinations are there for the first five balls?
MORE COUNTING EXAMPLES:
1.
Miss Murphy wants to seat 12 of her students in a row for a
class picture. How many different seating arrangements are
there?
2.
How many different 12-person juries can be chosen from a pool
of 20 jurors?
3.
Using the digits 1, 3, 5, 7, and 9, how many different 3-digit
numbers can be made if repetition of the digits is allowed?
4.
At Fredrico’s Pizza, they offer 10 different choices of toppings.
How many different 3-topping pizza’s could be made?
5.
A ship carries 10 different signal flags. If each possible
ordering of 4 of these flags connotes a specific message, how
many signals can be sent with these flags, taken 4 at a time?
6.
A student must answer 7 out of 10 questions correct on a test.
In how many ways could she do this?
7.
The final five Miss America Pageant contestants are to perform
their talents in an order chosen randomly. How many such
orderings are possible?
8.
A combination lock has a dial that can be set to any of the
numbers 1-50. How many different three-number combinations
are possible on this lock if no number can be repeated in the
combination?
How many different three-number combinations are possible on
this lock if the numbers can be repeated in the combination?
PROBABILITY EXAMPLES
(IN WHICH THE COUNTING IS A BIT TOUGHER)
1.
What is the probability of correctly choosing 5 numbers that
match five randomly selected balls when the balls are
numbered 1-39?
2.
What is the probability of guessing the top three winners, in
the correct order, from a group of 6 finalists in a spelling
bee?
3.
What is the probability of guessing the top three winners, in
any order, from a group of 6 finalists in a spelling bee?
4.
In Utah, license plates have 3 letters followed by three singledigit numbers. If Edwardo’s initials are EEM, what is the
probability that his licence plate will have his initials on it (in
order) if a licence plate is assigned to him at random?
5.
When five dice are rolled, what is the probability that all five
dice will show the same number?
6.
If you have a 4 digit password (where no digit may be used
more than once in the password), what is the probability that
I could guess the password in 5 tries or less?
7.
If a monkey randomly hits 4 keys on a 26-key typewriter
(one key for each letter of the alphabet), what is the
probability that it would type the word ‘MATH’?
8.
Suppose that you are dealt five cards from a shuffled
standard deck of 52 cards.
What is the probability of
being dealt a royal flush?
(A royal flush is A, K, Q, J,
10 all of the same suit)
What is the probability of
being dealt four of a kind?
(Four cards of the same
denomination and any other
card)
What is the probability that your hand contains no hearts?
What is the probability that your hand contains at least one
ace?