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 ofan experiment or observation.

EVENT An event consists of one or moreoutcomes that share a property ofinterest.

PROBABILITY The probability of an event, written,P(event), is a number that tells us howlikely it is that the event will occur. An event that is impossible has aprobability of 0.An event that is SURE to happen has aprobability of 1.All other events have a probabilitybetween 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, thenumber of outcomes that correspond to the event ofinterest.

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 52cards, what is P(a club is drawn)?

4. If I draw a marble fro a bag containing 3 red marbles, 2 bluemarbles, 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, allthree children are girls? (Assume that having a girl and having aboy 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 correspondingto 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 areflipped.

5. If the probability of a certain event is 2/5, what are the odds of thatevent?

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 theprobability that the card drawn is red or blue.

2. If a card is drawn from a standard deck of 52cards, find the probability that the card is an ace ora diamond.

3. If two dice are rolled, find the probability that thesum is 6 or doubles are rolled.

4. If two dice are rolled, find the probability that thesum is 3 or the sum is 9.

“AND” EXAMPLES:

1. What is the probability of drawing 2 Snickers barsin a row from a bag containing 6 Milky Ways and 3Snickers?

2. What is the probability of drawing 3 Snickers barsin a row from the same bag?

3. What is the probability of rolling three 6's in threerolls of a die?

“OR” AND “AND” PROBABILITY EXAMPLES:

1. Find the probability that 3 randomly selected people all have telephone numbersending 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 bothwrenches moderately often will break both during normal use?

5. When two dice are tossed, find the probability that either doubles are rolled or thesum 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 deckof 52 cards.

11. Three soccer teams will finish first, second and third in a field of 12 evenly matchedteams, 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 theprobability that there is at least oneother person in the room who has thesame birthday as you?

2. In a room of 30 people, whatis the probability that thereare at least two people in theroom who share the samebirthday?

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 theexperiment multiplied by their correspondingprobabilities.

1. Suppose someone offers you 10 to 1 odds thatyou will not get three heads when you toss acoin 3 times. That is, you lose $1 if you fail, andyou win $10 if you succeed. Is that a good betfor you? (What is the expected value of the bet).

2. Usually in roulette, betting on a red is a 1 to 1bet. That is, if you win, you gain the amount youbet; 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 policyholders will file a $5000 claim, 1 in 100 a$10,000 claim, and 1 in 200 a $30,000 claim in agiven year. Find the expected value to thecompany per policy sold.

HOUSE EDGE

Casinos make money because the expected value inevery game is negative (for the players.)

The house edge is the amount the casino can expect toearn per dollar that is bet.

1. Averaged over all games and all bets being played, aparticular casino has a house edge of 0.08 (8 centsper dollar gambled). If a total of $150 million is bet inthe casino over the course of a year, what is thecasino’s total profit?

2. A casino card game takes $4 bets. Players have a30% chance of winning (in which case they win backtheir bet plus $8), and they have a 70 % chance oflosing their bets. What is the house edge?

3. In a certain casino, the house wins on blackjack tableswith a probability of .507. All bets at blackjack are 1 to1. That is, if you win, you gain the amount you bet; ifyou lose, you lose the money you bet. What is thehouse edge in blackjack?

4. A roulette wheel has 18 black numbers, 18 rednumbers, and the numbers 0 and 00 in green. Usuallyin roulette, betting on a red is a 1 to 1 bet. What is thehouse 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 passwordsare 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),

then nr different arrangements are possible.

1. How many 4-character passwords are possible if thecharacters are chosen from the letters A-Z and thenumbers 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 byfour letters. How many such passwords are possible?

4. In Utah, license plates have 3 single-digit numbers followedby three letters. How many such license plates arepossible?

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 thedifferent 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 andthat 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 possibleorderings of the golfers are there?

COMBINATIONS

EXA: In how many ways can I pick two numbers from the numbers1, 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 thearrangement doesn’t matter, the number of possiblecombinations is:

COMBINATION EXAMPLES:

1. In how many different ways could a 3-person committee bechosen from a group of 10 people? From a group of 4 people?

2. How many different 5-card hands can be dealt from astandard deck of 52 cards?

3. Suppose that in POWERBALL, the first five balls are chosenfrom a set of balls labeled with the numbers 1-49. Howmany 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 aclass picture. How many different seating arrangements arethere?

2. How many different 12-person juries can be chosen from a poolof 20 jurors?

3. Using the digits 1, 3, 5, 7, and 9, how many different 3-digitnumbers 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 possibleordering of 4 of these flags connotes a specific message, howmany 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 performtheir talents in an order chosen randomly. How many suchorderings are possible?

8. A combination lock has a dial that can be set to any of thenumbers 1-50. How many different three-number combinationsare possible on this lock if no number can be repeated in thecombination?

How many different three-number combinations are possible onthis 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 thatmatch five randomly selected balls when the balls arenumbered 1-39?

2. What is the probability of guessing the top three winners, inthe correct order, from a group of 6 finalists in a spellingbee?

3. What is the probability of guessing the top three winners, inany order, from a group of 6 finalists in a spelling bee?

4. In Utah, license plates have 3 letters followed by three single-digit numbers. If Edwardo’s initials are EEM, what is theprobability that his licence plate will have his initials on it (inorder) if a licence plate is assigned to him at random?

5. When five dice are rolled, what is the probability that all fivedice will show the same number?

6. If you have a 4 digit password (where no digit may be usedmore than once in the password), what is the probability thatI 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 theprobability that it would type the word ‘MATH’?

8. Suppose that you are dealt five cards from a shuffledstandard 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 samedenomination and any other card)

What is the probability that your hand contains no hearts?

What is the probability that your hand contains at least oneace?