module 7
While theoretical probability is based on what may happen in theory, empirical probability is based
what actually happens in experiments or trials. This is why empirical probability is also called observational probability, as empirical probability is found through observations from collected data and recorded results. Empirical probability provides an estimate of the theoretical probability of an event from the relative frequency.
Theoretical probability is based on what
may happen in theory
20s 137 30s 115 40s 143 50s 165 What is the empirical probability that a person in their 30s is selected?
23/112 is Correct × There are 137+115+143+165=560 people. There are 115 people in their 30s. The empirical probability is 115560=23112.
What is the empirical probability of the spinner landing on green? Red 47 Blue 23 Yellow 59 Green 41
41/170 is Correct × There are 47+23+59+41=170 spins. There are 41 green spins. The empirical probability is 41170.
For the following questions determine the range of probability of the following scenarios and enter the letter that corresponds with your answer. a. 0% b. 40% to 60% c. 100% Use the information about playing cards below to answer the following questions. There are 52 playing cards in a complete set, also known as a deck. Within each deck, there are four categories, known as suits, with 13 cards in each suit. There are two black suits (clubs and spades) and two red suits (diamonds and hearts). Each suit has a card numbered two through ten, as well as a "jack", a "queen", a "king", and an "ace". The 13 cards of each suit are usually thought of in this ascending order. 5. The probability of drawing a red card from a deck of cards. Check 6. The probability of drawing a card numbered 2 through 10 or an ace, king, queen, or jack from a deck of cards. Check 7. The probability of drawing a #1 card from a deck of cards.
5. The probability of drawing a red card from a deck of cards The answer is 40% to 60%. Half of a deck of cards is red so the probability is between 40% and 60%. 6. The probability of drawing a card numbered 2 through 10 or an ace, king, queen, or jack from a deck of cards. The answer is 100%. Drawing any card from a deck of cards has a probability of 100%. 7. The probability of drawing a #1 card from a deck of cards .The answer is 0%. There are no #1 cards in a deck so there is 0% probability.
A survey on eye color was conducted. The results are 42 blue eyes, 58 brown eyes, and 20 other. What is the empirical probability of selecting a blue eyed person?
7/20 is Correct × There are 42+58+20=120 people. There are 42 people with blue eyes. The empirical probability is 42/120=7/20.
What is the empirical probability of rolling an odd number?
99/200 is Correct × There are 39+35+41+38+19+28=200 rolls. There are 39+41+19=99 rolls of an odd number. The empirical probability is 99/200.
Random experiments*
Accurately measuring relative frequency depends on having random experiments. 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. For example, you can flip a coin ten times, and each will be a random experiment. You do not know ahead of time whether you will get heads or tails. Getting heads or tails on one flip does not help you predict the result of the next flip.
Empty Set
An empty set* is a set that has no elements. There is nothing in the set; therefore, it is empty. This may seem odd, or even like it isn't a set at all, but an empty set is, in fact, a set. For example, let's say you wanted to list the days of the week that do not end in a y. There are none! Therefore, this is the empty set. In set notation, the empty set is written as a pair of brackets with nothing between them: {}
event
An 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.
Theoretical (Classical) Probability
An intuitive mathematical formula gives us a number that measures probability. If we are betting that a flipped coin will land on heads, we know there are two possibilities, heads and tails, and we win with only one of those possibilities (heads showing). It seems reasonable that the probability we will win this bet is 50% or 12 if expressed as a fraction.
Notation
As a reminder, sets are often listed between brackets, with commas separating each element in the set: { oak, juniper, elm, maple }
Quantitative Probability Examples As likely as unlikely
As likely as unlikely 40% to 60% probability A pregnant woman having a boy rather than a girl
Quantitative Probability Examples Certain
Certain 100% probability The probability that everyone alive today has a birthday
Calculating Relative Frequency
Empirical probability is estimated using the actual results of experiments or trials. Note that relative frequency only gives an estimate of the probability of an event occurring. As any sports fan knows, even if the same team (same players, same coach, etc.) played in the exact same league (same players, same coaches, same weather, etc.), their results two seasons in a row are unlikely to be the same. That is, for any two sets of experiments, the results vary each time. The first season this hypothetical team might win 25 out of 30 games; the next season, the team might only win 23 out of 30 games. Is the "true" probability that this team will win the next game 2530 or 2330 ? Either number is an estimate of the probability it will win the next game, but either number is more accurate than a 50% probability of winning that theoretical probability would give.
Example and Exercise relationship between theoretical and empirical
How does 50% compare to the relative frequency of heads in a set of data? See for yourself. The following Interactive Coin Toss activity will be used to answer the relative frequency questions below. Read the instructions and then click on the link below to run a couple of sets of experiments to answer the questions. First run an experiment by flipping a coin 10 times. Click on "No" for the question "Run Cumulative Stats" since we want each experiment to be isolated from the other experiments. Insert " 10 " into the "number of tosses" box. Click "Toss 'em" to run the experiment. To get the relative frequency, click "Ratio" on the side of the box to see the number of heads as a fraction. Now, run the experiment for 10 coin tosses, next for 100 coin tosses, and finally for 1000 coin tosses. For each trial record the decimal answer in the white box next to the corresponding question below."
For example, Bob has 5 different suits
If Bob likes the suits equally—that is, each morning he chooses one randomly—what is the probability he will choose to wear blue? P(Bob wearing a blue suit) = 3 blue suits5 suits total =3/5 The probability that Bob wears a blue suit is 3/5 . This can also be expressed as 0.6 or 60% .
empirical probability is based on .
experiments or trials.
Examples of Quantitative and Qualitative Probability Qualitative Description Quantitative Probability Examples impossible
Impossible 0% probability Probability that the day of the month will be 32
Subsets
In math and statistics, we often work with multiple sets at once. These sets can relate to one another. One such relation is known as a 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 .
In mathematics, a set is often, but not always, a collection of numbers:
In mathematics, a set is often, but not always, a collection of numbers:
Quantitative Probability Examples Likely
Likely More than 60% but less than 100% probability The probability that a child born in the United States will live to adulthood
Probability mathematically describes the likelihood of an uncertain event occurring.
Probability can be expressed as a decimal, fraction, or a percentage.
Probability and Subjectivity
Probability is a subjective area that often times requires judgement. In most all cases the extremes (impossible, certain) hold true. However in other cases, what one person might say is "unlikely," another person may find is "as unlikely as it is likely." Therefore, it is important to be mindful that while there is a measure of objectivity to many situations where probability is concerned, there are other situations where the likelihood of an outcome is often defined in the eye of the beholder.
Decimal Expression of Percentage
Probability percentages range strictly from 0% to 100% . It makes no sense mathematically to say there is 110% chance that X will occur—although people often say such things to emphasize their point. Therefore the decimal form of percentages range from 0 to 1 (remember that to obtain the decimal form of a percentage, you divide by 100 ).
By representing this set with a letter, we can more easily refer to this set.
Rather than listing all of the elements every time the set is referenced, we can simply say, " A has six elements" or "every number in A is an integer."
Relationship between Theoretical and Empirical Probabilities
Relationship between Theoretical and Empirical Probabilities We can discover the relationship between the theoretical and empirical probabilities by running experiments for which the theoretical probability is accurate, such as flipping a coin. The probability that someone will flip a "heads" in a coin toss is the number of desired outcomes (1, the head side) divided by the total number of possibilities (2, heads or tails). 12 is 0.5 or 50%
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 125 be most accurate?
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.
What is Probability?
Slide 1One familiar use of probability is the weather.End of First Slide Slide 2If you are hosting a barbecue in the summer, you might check ahead to see the probability it will rain on the Wednesday you have planned your party.End of Slide 2 Use tab to access slideshow controls Slide 3If the probability is 90% , you might be tempted to change the party date—or at least get ready to host it inside.
In a clinical trial, there are three different experimental dosages and a placebo. If a participant is equally likely to be assigned to any group, what is the probability he or she is assigned to the placebo group?
Since there are 3 treatment groups and 1 placebo group, there are a total of 4 groups. P(assigned to placebo group)=Number of placebo groups/Total number of groups=1/4 Note that 14 can be expressed as 0.25 or 25%/
The probability of an outdoor sporting event in Los Angeles being cancelled by bad weather.
The answer is unlikely. Los Angeles is known for having beautiful weather with fewer than 30 rainy days per year.
If you have a full, standard deck of cards, what is the probability of selecting a queen of hearts? a) 1/52 b) 1/26 c) 1/13 d) 1/4
The answer is a. There is a 152 probability of selecting a queen of hearts. There is one outcome that corresponds with the desired event (drawing the queen of hearts), out of 52 total possible outcomes (the total number of different cards that can be drawn)
For the following questions determine the probability of the following scenarios and enter the letter that corresponds with your answer. a. unlikely b. as likely as unlikely c. likely . An event with a probability of 9/20 .
The answer is as likely as unlikely. 9/20=0.45, which means the event is as likely as unlikely.
The probability of selecting a female rather than a male from a large elementary school.
The answer is as likely as unlikely. It is just as likely to select a female as it is to select a male.
When rolling a fair, six-sided die, what is the probability of rolling a 2 or a 3 ? a) 1/6 b) 1/3 c) 1/2 d) 2/3
The answer is b. There is a 26 probability of rolling a 2 or a 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. This fraction simplifies to 13 .
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? a) 1/7 b) 2/7 c) 3/7 d) 4/7
The answer is b. There is a 27 probability of selecting a green marble. There are two outcomes that correspond with the desired event (selecting either of the green marbles), out of seven total number of outcomes that can occur (the total number of different marbles that can be selected).
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? a) Theoretical Probability b) Empirical Probability c) The Law of Large Numbers d) All of the above
The answer is b. We are using the number of times an event occurred to estimate the probability, therefore this is an example of empirical probability.
Consider the following sets: A={42,23,11,35,73,97,32,26,41,85,48,61,15} B={12,23,95,73,27,9,26,43,82,18,63,15,99} The intersection of A and B is: a. {23,26,27,73} b. {15,23,61,73} c. {15,23,26,73} d. {15,26,48,73}
The answer is c. An intersection of two sets is a collection of the elements listed in both of the sets.
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? a. 14 b. 12 c. 34
The answer is c. Only 1 shirt is orange and three are not orange. So the probability is 34 , the number of desirable outcomes (not orange) divided by the total number of possibilities.
Which of the following is a subset of the following set? {22,23,31,45,53,67,72,76,81,85,88,91,95,100} a. {31,45,53,76,81,85,88,91} b. {22,23,45,67,72,92,95,100} c. {23,31,45,53,76,81,91,95,100} d. Both A and C e. Both B and C
The answer is d. Both of these sets are subsets.
How can probability be expressed in numeric terms? a. Fraction b. Decimal c. Percent d. All of the above
The answer is d. Probability can be expressed as a fraction, decimal, or percent to represent the chance of an event happening.
The Cardinals have won 100 of their 162 games. Using only this data, what is the probability that the Cardinals win a game? a) 100/162 b) 50/81 c) 62% d) All of the above.
The answer is d. This team has a 100162 probability of winning. 100162=5081=62% .
The probability of Christmas Day not being on December 25.
The answer is impossible. Christmas Day is always designated for December 25.
For the following questions determine the probability of the following scenarios and enter the letter that corresponds with your answer. a. unlikely b. as likely as unlikely c. likely An event with a probability of 78 .
The answer is likely. 78=0.875, which means the event is likely.
The probability of a student obtaining their high school diploma or GED.
The answer is likely. Most students end up with either their high school diploma or GED.
For the following questions determine the probability of the following scenarios and enter the letter that corresponds with your answer. a. unlikely b. as likely as unlikely c. likely An event with a probability of 1/4
The answer is unlikely. 14=0.25, which means the event is unlikely.
There are 48 dogs and 52 cats available for adoption at an event. If an animal is randomly chosen, what is the theoretical probability a cat will be selected? Enter the result as a decimal.
There are 48+52=100 pets. The theoretical probability is 52100=0.52.
A letter is chosen randomly from the word "worksheets". What is the theoretical probability a consonant is chosen? Enter the result as a decimal.
There are 10 letters and 7 consonants. The theoretical probability is 710=0.7.
An experiment is fair* if each outcome is
equally likely.
Intersection
The intersection* of two sets is a collection of the elements listed in both of the sets. For example: 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.
The Law of Large Numbers The Law of Large Numbers
The law of large numbers* states that as the number of trials increases, the relative frequency of an event will converge* on the theoretical probability. Simply put, the more times you flip a coin, the closer the relative frequency of heads will be to 50% . Notice a couple of things about the trials above. First, each of the results, except the first, had a relative frequency less than the expected 0.5 , but that is just random. Other trials would result in relative frequencies greater than the expected value. Second, although the difference between the relative frequencies and the expected value generally decreased, it did not do so uniformly. Again, each trial is random and the results cannot be guaranteed. Nevertheless it is true that as the number of flips gets larger, the relative frequency will get closer to 0.5 .
outcomes*
The possibilities of what can occur during an experiment—the results of the experiment—are called outcomes*. When rolling a die, 1 , 2 , 3 , 4 , 5 , and 6 are the possible outcomes.
A school contains 22 male teachers and 18 female teachers. The principal asks 1 teacher to come to the office. If each teacher is equally likely to be chosen, what is the theoretical probability a male teacher was selected? Enter the result as a decimal.
There are 22+18=40 teachers. The theoretical probability is 2240=0.55.
There is a 12 -sided number die with each face showing a number from 1 through 12 . What is the theoretical probability a double digit number was rolled? Enter the result as a decimal.
There are 3 possibilities, which are 10, 11, and 12. The theoretical probability is 312=0.25.
What is the empirical probability of rolling a number less than or equal to 3 ?
There are 39+35+41+38+19+28=200 rolls. There are 39+35+41=115 rolls less than or equal to 3. The empirical probability is 115/200=23/40
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.
There are 8 blue, 10 green, and 12 red marbles in a bag. One marble is selected from the bag at random. What is the theoretical probability the marble is red? Enter the result as a decimal.
There are 8+10+12=30 marbles in the bag. The theoretical probability is 1230=0.4.
When rolling a fair, six-sided die, what is the probability of rolling a 2 ? a) 1/6 b) 1/3 c) 2 d) 6
There is a 1/6 probability of rolling a 2 . There is one outcome that corresponds with the desired event, out of six total possible outcomes.
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? a. Only a few. Since the relative frequency can be used to substitute for the theoretical probability, a few surveys should give the researcher enough information to be able to make an accurate prediction. b. Some. Although relative frequency can be used to substitute for theoretical probability, the researcher will need the full range of information about screen time and ability to read social clues in order to get the full range of probabilities. c. Many. The researcher will need 100 surveys for the empirical probability estimated by relative frequency to equal the theoretical probability. d. 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.)
What is an example of a fair experiment? a. The chance of rain for the day b. The flipping of a coin c. Choosing a certain colored marble from a bag while looking into the bag d. A coin that is weighted to show "heads" more often than "tails."
This experiment has an equal probability for each possible outcome.
A union of two sets is a collection of the elements listed in both of the sets. True or False?
This is a false statement. A union of two sets is a collection of all of the elements listed in the sets.
An experiment is the procedure to test the occurrence of an event. True or False?
This is a false statement. An experiment is the procedure for which the probability of an event is calculated.
Consider the following two sets: A={14,15,16,17} B={11,12,13,14,16,17,18,19,20} A is a subset of B. True or False?
This is a false statement. By definition, to be a subset of a set, every element of the subset must be contained in the set. In this case, Set A contains 15 , which Set B does not contain. Therefore A is not a subset of set B .
In mathematics, a set is always a collection of numbers. True or False?
This is a false statement. In mathematics, a set is often, but not always, a collection of numbers.
After a certain number of trials the empirical probability of an outcome will equal the theoretical probability. True or False?
This is a false statement. The empirical probability will always be an estimate.
The probability of an event occurring can be greater than 100% . True or False?
This is a false statement. The probability of an event is always between 0% and 100% .
A die is rolled 10,000 times. We would most likely expect the relative frequency of rolling a " 1 " to converge on the value 16 . True or False?
This is a true statement. 10,000 tosses is a large number. By the law of large numbers we would expect the relative frequency to be close to 16 , the theoretical probability.
The possible results of an experiment are outcomes. True or False?
This is a true statement. Each of the possible results from an experiment is known as an outcome.
Sets
Throughout this course, we've been working with datasets. We've also seen sets as they relate to basic numeracy (the set of integers, the set of prime numbers, etc.). Formally, a set* is simply a collection of unique elements. For example, a set of tree species is: oak, juniper, elm, maple.
Empirical (Observational) Probability verses theoretical
To calculate theoretical probability, each outcome must be equally likely. The problem is that often in real life, outcomes are not equally likely. The classic example of a situation where theoretical probability does not provide much insight is when calculating the probability that a sports team will win a certain game. Theoretical probability would have to assume that each team is equally likely to win. But we know that in real life, often one team is far more likely to win than the other. We want a way to incorporate what we know about the teams to get a better sense of their chances.
Expressing Probability Probability, like other topics in math, brings with it a specialized vocabulary. First, read the quick example before the formal definitions are given:
Two friends bet on a roll of a die. If an even number, 2 , 4 , or 6 , appears, Jayne wins, while if an odd number, 1 , 3 , or 5 , appears, Stan wins. The chance of either person winning is 50% . Mathematicians and statisticians would call the roll of the die an "experiment." Each possible number that the die can land on is known as an "outcome." The possible outcomes of this experiment are rolling a 1 , 2 , 3 , 4 , 5 , or 6 . An "event" is one or more outcomes of an experiment. The die landing on an even number is an event, as is the die landing on an odd number. If all of the outcomes are equally likely, a statistician would call the die (or the experiment) "fair."
Quantitative Probability Examples unlikely
Unlikely More than 0% but less than 40% probability A snowstorm in Boston leaves more than 15 inches of snow
We often represent sets with a symbol or a letter, much like variables. For example, we can label a set as A:
We often represent sets with a symbol or a letter, much like variables. For example, we can label a set as A: A={−38,−1,1,35,63,94}
Introduction to Probability
What is probability? Put simply, it is the chance of an event occurring. Probability is often expressed in quantifiable terms as either a percentage* or a decimal: for example, the chance of a coin landing as heads is 50% . It is impossible to be certain you can correctly predict whether the coin will land heads or tails, which makes the act of flipping the coin a random event. We can however, calculate the probability* of the coin landing heads or tails.
Theoretical probability* is calculated
as the number of ways one particular event can occur in a random experiment, divided by the total number of possible outcomes:
Many probabilities are calculated by
creating a fraction (as we will learn on the next page). Therefore to calculate and express probabilities, you will need to be comfortable with percentages, decimals, and fractions and know how to convert among them. For a review of this topic, please see page 2.14.
Empirical probability*
gathers data by performing multiple experiments, or trials, and recording the results each time. For example, to have a better idea of the probability that a team will win a particular game, we naturally examine what other games it has won and against which teams. Empirical probability provides an estimate of the theoretical probability of an event from the relative frequency*, i.e how often the event occurs in the series of trials (or experiments) relative to the number of trials.
Relative frequency is
how often that event occurs in the trial (or experiment) relative to the outcomes of all the trials. That is, we take the number of times the event occurs divided by the total number of trials.
relative frequency*
how often the event occurs in the series of trials (or experiments) relative to the number of trials.
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).
Theoretical probability is calculated as
the number of ways one particular event can occur in a random experiment, divided by the total number of possible outcomes. It's important to note that in theoretical probability each outcome must be equally likely
A 0 probability means the event will definitely not occur, while 1 probability means the event will definitely occur. As the decimal values get closer to 1 ,
the probability of the event occurring increases. An event with a probability of 0.35 ( 35% ) is unlikely; an event with a probability of 0.65 ( 65% ) is likely; and an event with a probability of 0.9 ( 90% ) is very likely.
An 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.