C784 Module 7
Impossible
0% possibility Probability that the day of the month will be 32
The sum of all probabilities is
1 All whole fractions reduce to 1
Unlikely
1% to 30% probability A snowstorm in Boston leaves more than 15 inches of snow
You roll a six-sided die. What is the size of the sample space after one die roll?
6 The size of the sample space of the possible outcomes after one roll is 6. There are six possible outcomes.
Won Lost Won W, W W, L Lost L, W L, L What is the probability that you each win one round of rocks-paper-scissors-shoot, necessitating a third round?
1/2 here are four equally likely outcomes. Two of those outcomes involves you each winning one round of rocks-paper-scissors-shoot
6) You roll a six-sided die three times. What is the probability of rolling all three 1s?
1/216 The size of the sample space of the possible outcomes after three rolls is 216. These outcomes are equally likely. There is one outcome that results in rolling all three 1s.
If you have a full, standard deck of cards, what is the probability of selecting a queen of hearts?
1/52
Won Lost Won W, W W, L Lost L, W L, L While this table shows the sample space after two rounds, how large would the sample space be after just one round?
2 The sample space after one round would be 2.
Sally has 3 coats - one is blue, one is red, and one is green. She also has red and green gloves. Everyday she wears one coat and one set of gloves. Which of the following uses the General Addition Rule to calculate the probability that Sally wears green on any given day? (The fractions are unsimplified to make calculation easier.)
2/6 + 3/6 - 1/6 The General Addition Rule gives us a way of calculating the probability of two independent events. (RC, RG) (GC, RG) (BC, RG) (RC, GG) (GC, GG) (BC, GG) Notice that the probability of Sally wearing a green coat is 2 out of 6 cells in the table. The probability that she wears green gloves is 3 out of 6 cells in the table, but we have double counted wearing both a green coat and green gloves, so we must subtract the one cell in which she is wearing both green items.
You have a jar of 7 marbles: 1 red marble, 2 green marbles, and 4 blue marbles. Selecting a marble at random, what is the probability that you select a green marble?
2/7
5) You roll a six-sided die three times. What is the size of the sample space after those three rolls?
216 The size of the sample space of the possible outcomes after three rolls is 216. There are 216 possible outcomes.
Won Lost Won W, W W, L Lost L, W L, L While this table shows the sample space after two rounds, how large would the sample space be after three rounds, assuming all three rounds are played regardless of the outcome of the first two rounds?
8 The sample space after three rounds would be 8. WWW LLL WWL LLW WLW LWL WLL LWW
VENN DIAGRAM
A type of chart that illustrates how distinct sets, topics, or objects relate to one another. Uses circles to represent events
POPULATION
An entire pool from which a sample is drawn. Samples are used in statistics because of how difficult it can be to study an entire population
A six-sided die is showing a 5 and a 1 disjoint or non-disjoint
DISJOINT Only a 5 or a 1 can be showing on the die, not both
Disjoint Events: Dependent
Disjoint events are dependent. For example, If you know someone is not born on a Wednesday, there is a greater possibility, 1/6 he or she is born on a Friday. Because the occurrence of one affects the probability of the other, these events are dependent.
1. When researching or trying to learn something about a given population, the population and the sample should always be disjoint. True or False?
FALSE While a population and a sample can be disjoint, they shouldn't be if you're trying to use your sample to learn something about your population!
3. The only information you need to perform analysis in a two-way table can be provided by row entries. True or False?
FALSE You also need the totals in columns and rows to perform analysis.
3. In the problem above, making a puppet with a red body and making a puppet with a blue body are independent events. True or False?
FALSE You can only choose one color for the body of the puppet, so choosing a red-bodied puppet and choosing a blue-bodied puppet are disjoint events.
How can probability be expressed in numeric terms?
FRACTIONS DECIMALS PERCENTS
5. Events that do not influence each other are independent. True or False?
Independent events are events that do not affect each other.
INDEPENDENT EVENT
Independent events* are those that are not affected by other trials or events. For example, if you were to flip a coin once, that first result (either heads or tails)
Disjoint A and B (intersection)
It is relatively easy to calculate the probability of two or more disjoint events. If there is no overlap, there is no possibility of belonging to both sets. The probability that someone is born in both March and January, for example, is 0. This is true of the intersection of all disjoint events: If A and B are disjoint, P(A and B) = 0
A number shown on a die is even and a number is less than 3 disjoint or non-disjoint
NON DISJOINT 2 is even and 2 is less than 3, so these events are not disjoint
General Addition Rule for events that are disjoint
P(A or B) = P(A) + P(B) the general rule simplifies to P(A or B) = P(A) + P(B) since there is no area of intersection.
General Addition Rule events that are not disjoint*
P(A or B) = P(A) + P(B) - P(A and B)
In another game, you win if you roll a number less than 3 or if you roll an even number. What is your probability of winning this game?
P(A or B) = P(A) + P(B) - P(A and B). There are 2 numbers less than 3, so the probability of event A is 2/6 or 1/3 There are 3 numbers that are even, so P(B) = 1/2 But the number 2 is both less than 3 AND even, so P(A and B) = 1/6 The formula gives us 1/3 + 1/2 - 1/6 = 2/3
SUBSET
Set A is a subset* of set B, if every element in A is contained within B. For example: A = {1, 2, 3} B = {1, 2, 3, 4, 5} A is a subset of B, because every element in set A is contained within set B.
Disjoint A or B (union)
Since there is no overlap, the probability of being in either one or the other is the sum of their individual probabilities. What is the probability that someone is born in January or March, for example? 2/12 OR 1/6
2. A sample space S is comprised of 13 equally likely outcomes. Suppose event E contains 5 of those outcomes. The probability of E is 5/13 True or False?
TRUE Correct. This is a true statement. Each event has a 1⁄13 probability of occurring, so one of 5 occurring is 5/13
INTERSECTION
The intersection* of two sets is a collection of the elements listed in both of the sets. E = {0, 10, 100} F = {-2, -1, 0, 1, 2} The intersection of E and F is {0}, as 0 is the only element that appears in both sets.
OUTCOME
The possibilities of what can occur during an experiment—the results of the experiment— When rolling a die, 1, 2, 3, 4, 5, and 6 are the possible outcomes
P(E)
The probability of an event is represented by P(E) which means probability (P) of a certain event (E) occurring. If R = Rain and the weatherman says there is a 40 percent chance of rain, then P(R) = 0.40. (Notice that the probability is written in its decimal form).
4. Relative frequency helps determine if an event is independent. True or False?
The relative frequency is related to the independence of events.
what is the sample space for being born on a day of the week?
The sample space would be Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday
SAMPLE SPACE
The set of possible outcomes in an experiment The sample space of rolling a regular six-sided die is 1, 2, 3, 4, 5, and 6
You create a table to determine the size of the sample space of an experiment. What is the size of the sample space equal to?
The size of the sample space is equal to the number of cells in the table (not counting any labels of the rows or columns).
UNION
The union* of two sets is a collection of all of the elements listed in the sets For example: C = {2, 4, 6} D = {1, 3, 5} The union of C and D is {1, 2, 3, 4, 5, 6}, as those are all of the elements that appear in the sets.
UNIVERSE
The universe is the entire category being considered. It encompasses both A and B and outcomes that are not in either category. In our example of Bob's shoes and suits, the universe would be Bob's outfits, the ensemble of suits and shoes. These outfits would not include shirts, ties, or socks! Every cell in the table would be included in the universe.
"NOT"
The word "not" before a category is everything outside that category but within the universe.
SET
a collection of unique elements For example, a set of tree species is: oak, juniper, elm, maple.
SAMPLE
a group of people chosen to represent a larger population
RANDOM EXPERIMENTS
are trials in which the outcome is not known ahead of time and the result does not depend on the results of other trials.
THE LAW OF LARGE NUMBERS
as an experiment is repeated again and again, the empiracal probability will get closer and closer to the theoretical probability
INDEPENDENT
events, where the occurrence of one does not affect the probability that the other event will occur
ADDITION RULE OF PROBABILITY
for disjointed events A and B, P(A or B) = P(A) + P(B)
AND
only that specific choice ("A" and "B")
PERCENTAGE
parts of a whole
Decimal Expression of Percentage
probability percentages range strictly from 0% to 100% Therefore the decimal form of percentages range from 0 to 1 (remember to obtain the decimal form of a percentage, you divide by 100)
OR
the choices and the mixture of the choices (A), (B), (A,B)
COMPLEMENT
the opposite of an event happening (i.e. the event not happening) For example, the complement of flipping heads is flipping tails
Independent events
the probability *you are born in march, given that your spouse is born in march *flipping tails given that you just flipped heads
4. What is the sample space for choosing an ice cream flavor (chocolate, strawberry, vanilla) and sauce (chocolate, caramel, or none)?
The answer is 9. From the list that was constructed, there are 9 outcomes. Vanilla, Chocolate Vanilla, Caramel Vanilla, None Chocolate, Chocolate Chocolate, Caramel Chocolate, None Strawberry, Chocolate Strawberry, Caramel Strawberry, None
You examine a deck of cards to determine the probability of selecting a certain group of cards. What is this an example of? Theoretical probability? Empirical Probability? The law of large Numbers?
Theoretical probability We are using the number of times an outcome would occur, and they are all equally likely, this is an example of theoretical probability.
When rolling a fair, six-sided die, what is the probability of rolling a 2 or a 3?
1/3 There are two outcomes that correspond with the desired event (rolling a two or a three), out of six total number of outcomes that can occur. 2/6 This fraction simplifies TO 1/3
You roll a six-sided die twice. What is the probability of rolling two sixes?
1/36 The size of the sample space of the possible outcomes after two rolls is 36. These outcomes are equally likely. There is one outcome that results in two sixes
Won Lost Won W, W W, L Lost L, W L, L What is the probability that you lost both rounds of rocks-paper-scissors-shoot?
1/4 There are four equally likely outcomes. One of those outcomes involves losing both rounds
Won Lost Won W, W W, L Lost L, W L, L What is the probability that you won both rounds of rocks-paper-scissors-shoot?
1/4 There are four equally likely outcomes. One of those outcomes involves winning both rounds
4) You roll a six-sided die twice. What is the probability of rolling the same number both times?
1/6 The size of the sample space of the possible outcomes after two rolls is 36. These outcomes are equally likely. There are six outcomes that results in rolling the same number both times [(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)].
When rolling a fair, six-sided die, what is the probability of rolling a 2?
1/6 There is one outcome that corresponds with the desired event, out of six total possible outcomes
1. A coin is flipped three times. What is the probability of flipping three heads in a row (do NOT construct a tree; instead, use the multiplication principle)?
1/8 There are 2 possibilities for each coin toss. The probabilities are 1/2 * 1/2 *1/2 = 1/8
2. A puppet-making booth has the choice of 3 different color bodies and 3 types of head. What is the probability of making one particular type of puppet?
1/9 There are 3 body types and 3 types of head, there are 9 different kinds of puppets. 1/3 *1/3 *1/3 = 1/9
Certain
100% probability The probability that everyone alive today has a birthday
The Cardinals have won 100 of their 162 games. Using only this data, what is the probability that the Cardinals win a game?
100/162 50/81 62%
Jeni commutes along the same route as Ricky, only at different hours. She is caught in traffic on the commute home if it is raining AND there is a lane closure; otherwise there is no traffic. Like Ricky, Jeni experiences rain on her commute 30% of the time and a lane closure 20% of the time. What is the probability that she is caught in traffic on any given day?
This is an "and" problem, the probability that it is raining AND there is a lane closure. P(A) = 0.3 P(B) = 0.2 P(A and B) = 0.3 x 0.2 6% of the time, she will be caught in traffic
Kiki the dog has a yellow, blue, green, and orange shirt. If her owner reaches into the drawer and picks one out at random, what is the probability the shirt is NOT orange?
3/4
Won Lost Won W, W W, L Lost L, W L, L What is the probability that you do NOT win both rounds of rocks-paper-scissors-shoot?
3/4 here are four equally likely outcomes. Three of those outcomes do NOT involve you winning both rounds
You roll a six-sided die twice. What is the size of the sample space after those two rolls?
36 The size of the sample space of the possible outcomes after two rolls is 36. There are 36 possible outcomes.
As likely as unlikely
40% to 60% probability A pregnant woman having a boy rather than a girl
Likely
70% to 99% probability The probability that a child born in the United States will live to adulthood
EXPERIMENT
is the procedure, or situation, for which the probability is being calculated. So the roll of a die can be an experiment, a coin flip can be an experiment, or even the weather on a particular Wednesday
PROBABILITY
it is the chance of an event occurring
EMPIRICAL (OBSERVATIONAL) PROBABILITY
Based on what ACTUALLY HAPPENS --estimated by relative frequency Relative frequency= # of times the event occured/ Total # of trials
What is the sample space for flipping 3 coins?
The answer is 8. From the list constructed below, there are 8 outcomes. Heads, Heads, Heads Heads, Heads, Tails Heads, Tails, Heads Heads, Tails, Tails Tails, Tails, Tails Tails, Tails, Heads Tails, Heads, Tails Tails, Heads, Heads
If the probability of her having time by getting the earlier train is 75% and the probability of the shop still having chocolate donuts is 50%, what is the probability that she gets her chocolate donut?
We are looking for an intersection here, an "and" problem. P(A and B) = P(A) x P(B) = 0.75 x 0.5
RELATIVE FREQUENCY
i.e how often the event occurs in the series of trials (or experiments) relative to the number of trials. Relative frequency= # of times the event occured/ Total # of trials
FAIR
if each outcome is equally likely.
EMPTY SET
is a set that has no elements
EVENT
is comprised of one or more outcomes. The die landing on an even number is an event. The die landing on 1 is also an event
SAMPLE SIZE
is the number of different outcomes the sample size of rolling a six-sided die is 6.
dependent Events
the probability *of getting lung cancer given a history of smoking *that your car is red, given that your previous car was red *your good at sports given that one parent was good at sports *picking a card that is a heart, while holding 3 hearts in your hand
What is an example of a fair experiment?
flipping a coin This experiment has an equal probability for each possible outcome.
A researcher is testing the hypothesis that more screen time decreases a person's ability to read social clues. The researcher is using surveys to gather information both about a person's screen time and about his or her ability to read social clues in order to be able to predict how well a person with a certain amount of screen time will be able to read social clues. How many surveys should the researcher collect?
As many as possible. The law of large numbers holds that the more trials, the closer the empirical data comes to estimating the true probability. There is no way to calculate the theoretical probability in a case like this. The more data the researcher gathers the clearer an idea he or she will have about the extent of its influence (if any.)
COMPLEMENTARY EVENT
Complementary events are those that do not have any common outcomes, and the union is the whole universe If there are outcomes in the universe that are not in either event, then the two events are not complementary The events must be properly defined in order to have complementary events For example, "rolling a die and getting an even number" would not be a complete instance of complementary events (as this is only one event). However, "rolling a die and getting an even number" and "rolling the die and getting an odd number" are complementary events; the events are clearly defined.
Example: An Event That is Not Independent
In a candy container, there are five white chocolates and ten dark chocolates. If you pick a chocolate at random out of the jar, the probability of what you will get on the next pick is affected. If you pick a dark chocolate on your first pick, there will be fewer dark chocolates the next time you pick, lowering the probability you get a dark one on the second draw
You examine the number of times a player hit a home run this season, divided by the number of at-bats the player had. What is this an example of? Theoretical probability? Empirical Probability? The law of large Numbers?
Empirical Probability We are using the number of times an event occurred to estimate the probability, therefore this is an example of empirical probability.
5. The frequency table does not need labels for rows or columns. True or False?
FALSE A frequency table needs labels for rows and columns to display the data accurately.
4. A tree diagram is the only method to determine the probability of an event. True or False?
FALSE A list, table, tree diagram, and other methods can help determine the probability of an event.
. An experiment is the procedure to test the occurrence of an event. True or False?
FALSE An experiment is the procedure for which the probability of an event is calculated.
4. If 2 events can occur at the same time, they are called disjoint. True or False?
FALSE Disjoint events are 2 events that cannot occur at the same time.
After a certain number of trials the empirical probability of an outcome will equal the theoretical probability. True or False?
FALSE The empirical probability will always be an estimate.
3. Using a tree diagram can result in probability greater than 1. True or False?
FALSE The probability of an event can never be greater than 1.
All events in the sample space have the same relative frequency. True or False?
FALSE The relative frequency of an event is how often it happens in practice. We do not know if people who do not have inherited risk factors for heart disease decrease their risk of developing heart-related diseases by eating healthy until we gather data. It is reasonable to assume, however, that the relative frequency of different eating habits (healthy versus not-healthy) might not be equal.
The sample space and sample size represent the same thing. True or False?
FALSE The sample space is a set of outcomes while the sample size is the number of different outcomes.
4. Complementary events occur whenever there are two events. True or False?
FALSE The two events must be defined events.
2. If two events have no common outcomes, they are complementary. True of False?
FALSE Two events with no common outcomes are complementary only if their union is the whole universe
MULTIPLICATION RULE FOR INDEPENDENT EVENTS
If two events are independent P(A and B) = P(A) x P(B) For example, the probability of rolling a one with a six-sided die is 1/6. When rolling two dice with six sides each there are 1/6 X 1/6 = 1/36 probability of rolling double ones. You can verify this probability on the table below. Notice there is only one cell with double ones and there are 36 (6 × 6) cells in total. Therefore, the probability of double ones is 1/36 Likewise, the probability of rolling triple ones when rolling 3 dice is 1/6 X 1/6 X 1/6 = 1/216
DISJOINT
If two events cannot both occur at the same time, they are called disjoint Examples of Disjoint Statements I was born in January. I was born in March. I was born in May.
In a certain game, you roll a fair, six-sided die once. You win if you roll a 3 or a 4. What is the probability of winning this game?
These are disjoint events. P(A or B) = 1/6 + 1/6 33% chance of winning
Joannie chooses a dessert recipe to bake for her book club meeting from the 25 dessert recipes she has in her recipe box. To predict the likelihood of her making a chocolate chip cookie recipe next book club meeting, would the theoretical probability 1/25 be most accurate?
NO She likely prefers some recipes over others or more often has the ingredients at hand. To help predict the future, an empirical probability will work best.
A 3 year old weighs at least 25 lbs and the same 3 year old weighs 27 lbs. disjoint or non-disjoint
NON DISJOINT a3 year old can weigh 27 lb and can weigh over 25 lbs
If the probability of a sunny day is 0.7 and the probability of the date on the calendar being an odd number is 0.5, what is the probability of the calendar date being odd or it being a sunny day?
P(A or B) = P(A) + P(B) - P(A and B) Since "calendar date being an odd number" and "being a sunny day" are independent events, P(A and B) = 0.7 * 0.5 = 0.35. Calendar date being an odd number or being a sunny day = 0.7 + 0.5 - 0.35 = 0.85 85% chance of it being sunny or being an odd numbered day on the calendar.
the probability of the complement.
P(A) + P(not A) = 1 1 - P(A) = P(not A) 1 - P(not A) = P(A)
A die is rolled 10,000 times. We would most likely expect the relative frequency of rolling a "1" to converge on the value 1/6 True or False?
TRUE 10,000 tosses is a large number. By the law of large numbers we would expect the relative frequency to be close to 1/6, the theoretical probability.
1. A frequency table consists of two categorical variables. True or False?
TRUE A frequency (or two-way) table contains two different categorical variables.
Any experiment or trial will have only one sample space. True or False?
TRUE A sample space is the set of outcomes for one particular experiment, so any experiment has one unique sample space.
2 A frequency table is read by the row and column entries. True or False?
TRUE Each entry is read by the row and column entry.
In a tree diagram showing the sample space of an experiment, each new option increases the sample size. True or False?
TRUE Each new option will be added to each of the previous options, increasing the size of the sample space dramatically.
The possible results of an experiment are outcomes. True or False?
TRUE Each of the possible results from an experiment is known as an outcome.
1. The sample space can have equally likely outcomes. True or False?
TRUE Each outcome in a sample space can have the same chance of occurring.
5. Finding the probability of events, given all outcomes are equally likely, has limited applications. True or False?
TRUE Most probabilities are calculated when there are not equally likely outcomes.
A list, table, or tree diagram will show the exact same sample space. True or False?
TRUE No matter the method used, each technique will have the same sample size.
1. The complement of an event occurring is the event not occurring. True or False?
TRUE The opposite of an event happening is the event not occuring.
2. The probability using sample spaces with equally likely outcomes is the number of outcomes in the event divided by the sample size. True or false?
TRUE The probability is the number of outcomes in the event divided by the sample size.
5. The sum of all probabilities of individual outcomes in the sample space must equal 1. True or False?
TRUE The sum of all probabilities of individual outcomes equals 1.
3. If the probability of the event given is 0, then the complementary event will have probability equal to 1. True or False?
TRUE The sum of the probabilities of the event and its complement is 1, so if the probability of the given event is zero, its complement has probability equal to 1.
5. If two events are disjoint there is no intersection between the events. True or False?
TRUE There are no common elements between 2 disjoint events.
3. The addition rule for disjoint events would yield the same result as the general formula for theoretical probability, counting the ways of getting a certain result. True or False?
TRUE We have done problems both ways in this module. We add the individual probabilities for the disjoint events of, say, being born in February OR being born in April. Or we count up all the "ways" of being born in either February or April and put that number over the total size of the sample space. Either method for this problem would yield 2/12 = 1/6
It is important to remember that the probabilities of all the outcomes in the sample space always sum to 1.
That is, there is a 100 percent chance of one of the outcomes in the sample space happening, since the sample space, by definition, contains all possible outcomes.
What is the sample space for rolling 1 dice and flipping 1 coin?
The answer is 12. From the list that was constructed, there are 12 outcomes. 1, Heads 1, Tails 2, Heads 2, Tails 3, Heads 3, Tails 4, Heads 4, Tails 5, Heads 5, Tails 6, Heads 6, Tails
3. What is the sample space for the order in which marbles are drawn out of a bag that contains 1 red, 1 yellow, and 1 green marble?
The answer is 6. From the list that was constructed, there are 6 outcomes. Red, Yellow, Green Red, Green, Yellow Yellow, Red, Green Yellow, Green, Red Green, Red, Yellow Green, Yellow, Red
THEORETICAL (CLASSICAL) PROBABILITY
based on what MAY happen or on a theory is calculated as the number of ways one particular event can occur in a random experiment, divided by the total number of possible outcomes: EACH OUTCOME MUST BE EQUALLY LIKELY
The probability of an event occurring can be greater than 100%. True or False?
false The probability of an event is always between 0% and 100%.
