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: STEP 2: STEP 3: Count the total number of possible outcomes. Count the number or favorable outcomes that is, the number of outcomes that correspond to the event of interest. 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), then n r 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?