Mutually exclusive events - two events are mutually exclusive if they cannot happen at the same time i.e. they have no outcomes in common. A single coin is tossed. There are two events: observing a head or observing a tail. Observing a head excludes the possibility of observing a tail. Two events are non-mutually exclusive if they have one or more outcomes in common. A single die is tossed. Event A: observe a four. Event B: observe an even number. These events have one outcome in common since an "even number" consists of the numbers 2, 4, and 6.
Probability - a number between O and I that measures how likely an event is to happen over the long run. A probability of I means an event happens with certainty and a probability of O means it is impossible for an event to happen.
Arithmetic mean of a set of outcomes is usually called the average value of these outcomes. To
find the mean of the numbers 1,8,6,4,7, we add the numbers together-26-and divide by 5 = 5.2.
Variability shows how concentrated or spread out the outcomes are around the arithmetic mean.
Expectation is the average value we expect to observe if we perform a large number of experiments. Also called expected value- the probability-weighted sum of all possible outcomes.
Population - the total number of something - outcomes, objects, events, etc. It is a group that has at least one characteristic in common from which our sample of data is selected.
278
Sample - a representative and randomly taken part of the population that is studied in order to draw conclusions about the population. The larger the sample, the better our estimate of the probability. But observe that what is essential is the absolute size of the sample (e.g. the number of people who've been asked), not its proportion of the population. A random sample of 3,000 from the entire U.S. is more predictive than a sample of 40 from a particular university. A random sample is one in which every item in the population is equally likely to be chosen.
How do we decide the probability of an event?
The laws of probability tell us what is likely to happen over a large number of trials. This means we can expect to make reasonable predictions what on average are likely to happen over the long run, but we can't predict the outcome of a particular event.
There are three ways to measure probability: the logical way, the relative frequency way, and the subjective way.
The logical way
The logical way can be used in situations where we know the number of outcomes and where all these outcomes are equally likely. For example in games of chance, we find the probability by dividing the number of outcomes that are favorable to the event (yield the outcome we look for) with the total number of possible outcomes. Observe that this definition can only be applied if we can analyze a situation into equally likely outcomes.
What is the probability that we observe one head when tossing a coin? The number of equally likely favorable outcomes is one (head) and the total number of possible equally likely outcomes is two (head and tail) and therefore the probability is½ or 50%.
Relative frequency
When an experiment can be repeated many times, probability is the proportion of times the event happened in relation to an infinite number of experiments. In most cases we don't know the probability of an event. Why? Because we can't know all the outcomes. We must then try to estimate the long-term relative frequency by performing experiments or by finding representative information about how often an event happened in the past.
By representative we mean that the information must be based on the relative frequency of historical data over a large number of independent experiments or observations of the same reference class under essentially the same conditions. The reference class is the one for which the distribution of outcomes is known or can be reasonable estimated. The more relevant cases we examine, the better our chance is to estimate correctly the underlying probability.
Conduct an experiment to test how likely it is that you toss head. Toss a coin 1000 times and observe what happens. Jfyou got heads 400 times the relative frequency or the fraction between the number of tossed heads (number of times the events happened) and the total number of tosses (number of experiments) is 400/1000. Toss a coin 2000 times and observe what happens. If you got heads 900 times the relative frequency is 900/2000. The more tosses, the less the difference will be between an events theoretical probability and the relative frequency with which it happens. In this case it will move towards the value ½.
279
What is the frequency of losses? How are losses distributed over time? What is their magnitude? Insurance companies use relative frequencies. They base their premiums on an estimate of how likely it is that a specific event that cause them to pay happen. If they assume the past is a representative guide for the future they try to find out the relative frequency of a specific "accident" by observing past frequencies of specific accidents.
Suppose the probability that a given house will burn is 0.3%. This means that the insurance company found historical data and other indicators about a large number of houses (for example the reference class is "50 years of data of fires in a given area'') and discovered that, in the past, 3 out of 1,000 houses in a given area will burn. It also means that, assuming there are no changes in the causes of these fires, we can predict approximately the same proportion of fires in the foture. An insurance company knows that in a given year a certain percentage of their policyholders will have an accident. They don't know which ones, but they diversify their risk by insuring many individuals. What is unpredictable for one person can be predictable for a large population. But they must make sure the events are independent and that not a single event or a confluence ofindependent events affects multiple policyholders causing the insurer to pay out on many claims at one time. For example, an insurance company who provides fire insurance to a number of buildings in a single block may face ruin if a large fire happens.
Subjective probability
If an experiment is not repeatable or when there is no representative historical relative frequency, or comparable data, the probability is then a measure of our personal degree of belief in the likelihood of an event happening. We have to make a subjective evaluation or a personal estimation using whatever information is available. But we can't just assign any number to events. They must agree with the rules of probability.
A New York Rangers supporter might say, "I believe that the NY Rangers have a 90% chance of winning their next match since they have been playing so well. "
Rules of probability
When two events are independent (no event can influence the probability of the other), the probability that they both will happen is the product of their individual probabilities. We can write this as: Both A and B will happen= P(A) x P(B).
A company has two independent manufacturing processes. In process one, 5% of the produced items are defective, and in process two, 3 %. If we pick one item from each process, how likely is it that both items are defective? 0.15% (0.05 x 0.03).
This rule is changed if the events are dependent. In many situations the probability of an event depends on the outcome of some other event. Events are often related in a way so that if one event happens, it makes the other event more or less likely to happen. For example, if we toss a die and event A is: observe an even number, and event B: observe a number less than 4, then given we know B has happened, the probability is 1/3. This is called a conditional probability or the probability that an event happen given that some other event has happened. Conditional probabilities involve dependent events. The conditional probability of A given B is 1/3 since we know B was either 1,2 or 3 and only 2 implies event A.
280
What is the probability that there are two boys in a two-child family given that there is at least one boy? Ask: What can happen or what are the number of outcomes that are equally likely to happen? Boy/Boy- Girl/Girl - Boy/Girl - Girl/Boy. Since we already know that there is "at least one boy'; we can rule out scenario "Girl/Girl" Th
e probability is therefore 113 or 33%.
What is the probability that there are two boys in a two-child family given that the first-born is a boy? The number of outcomes that are equally likely to happen are: Boy/Boy - Girl/Girl - Boy/Girl - Girl/Boy. Since we already know that the older child is a boy, we can rule out scenario GB and GG. The probability is 50%.
A problem in conditional probability that has caused many mathematics professors problems, is the Monty Hall Dilemma. The columnist Marilyn vos Savant (Parade 1990, Sept 9, p. 13.) asked the following problem:
"Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car; behind the others, goats. You pick a door, say No. l, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to switch your choice?"
How would you answer? Assume we always have the opportunity to switch. Make a table of possible outcomes and test in how many outcomes it pays to switch doors.
Door 1
Door2
Door3
Car
Goat
Goat
Goat
Car
Goat
Goat
Goat
Car
Assume you choose door number 1. What are then the consequences if the car is in door 1,
2 or 3?
Car behind door
Host 012ens door
You switch
You don't switch
1
2
Lose
Win
2
3
Win
Lose
3
2
Win
Lose
2/3
1/3
We should always switch doors since we win 2/3 of the time. They key to this problem is that we know ahead of the game (conditional) that the host knows what is behind each door and always opens a door with a goat behind it.
When two events are mutually exclusive (can't both happen at the same time), the probability that one or the other will happen is the sum of their respective probabilities. We can write this as: Either A or B will happen = P(A) + P(B).
What is the probability that we get either a two or a four when a single die is tossed once? There are 6 outcomes and the two events ("getting a two" and ''getting a four") have no outcomes in common. We can't get both a two and a four on the same toss. How many favorable outcomes are
281
there or in how many ways can we get a two? In one out of six. In how many ways can we get a four? In one out of six. The probability that we get a four or a six is 1/6 + 1/6 = 33%.
When two events are non-mutually exclusive (both can happen at the same time), the probability that at least one of them will happen is equal to the sum of the probabilities of the two events minus the probability that both events happen. We can write this complementary rule as: P(A) + P(B) - P(A and B).
Assume that the probability a Los Angeles teenager ownsa surfing board is 25%, a bicycle 85%, and owning both 20%. If we randomly choose a Los Angeles teenager, the probability that he or she owns a surfing board or a bicycle is (0.25 + 0.85) - 0.20 = 90%. These events had two outcomes in common since the teenager could have both a surfing board and a bicycle.
Sometimes it is easier to deal with problems if we turn them backwards. The probability of an event not happening is 1 minus the probability it will happen. If the probability of an event A is 30%, then the probability that the event won't happen is 70% because "not event A" is the complement of the event A. The sum of probability of an event happening and probability of the same event not happening is always 1.
What is the probability that we get at least one six in four rolls of a single die? We turn the question around and calculate the probability of "not getting any sixes in four rolls ofa die''. There are four independent events - not getting a six on the first throw... second... The probability of each one is 516 since there are five outcomes (1,2,3,4,5) which results in the event "no six" and each is independent of what happened before. This means that the probability of not getting any sixes are 516 x 516 x 516 x 516 or 48.2%. Therefore "the probability ofgetting at least one six" is 1
- 0.482 or 51.8%.
Counting possible outcomes
The multiplication principle says that if one event can happen in "n" different ways, and a second event can happen independent of the first in "m" different ways, the two events can happen in nm different ways.
Suppose there are 4 different flights between Los Angeles and New York, 3 between New York and Boston and 5 between Boston and Bermuda. The number of itineraries assuming we can connect between any of the stated flights are 4 x 3 x 5 = 60.
Permutations or rearrangements mean the different ways we can order or arrange a number of objects.
We have 3 hats to choose from - one black, one white and one brown. In how many ways can we arrange them if the order white, black and brown is different from the order black, white and brown? This is the same as asking how many permutations there are with three hats, taken three at a time. We can arrange the hats in 6 ways: Black-White-Brown, Black-Brown-White, White Black-Brown, White-Brown-Black, Brown-White-Black, Brown-Black-White.
Another way to think about this: We have three boxes in a row where we put a different hat in each box. We can fill the first box in three ways, since we can choose between all three hats. We can then fill the second box in two ways, since we now can choose between only two hats. We can fill
282
the third box in only one way, since we have only one hat left. This means we can fill the box in 3 x 2 x 1 = 6 ways.
Another way to write this is 3!If we haven (6) boxes and can choose from all of them, there are n (6) choices. Then we are left with n-1 (5) choices for box number two, n-2 (4) choices for box number three and so on. The number of permutations of n boxes is n!. What n! - Factorial- means is the product of all numbers from 1 ton.
Suppose we havea dinner in our home with 12 people sitting around a table. How many seating arrangements are possible? The first person that enters the room can choose between twelve chairs, the second between eleven chairs and so on, meaning there are 12! or 479,001,600 different seating arrangements.
The number of ways we can arrange r objects from a group of n objects is called a permutation of n objects taken rat a time and is defined as n! / (n-r)!
A safe has 100 digits. To open the safe a burglar needs to pick the correct 3 different numbers. Is it likely? The number ofpermutations or ways ofarranging 3 digits from 100 digits is 970,200
(100!/(J 00-3J!). If every permutation takes the burglar 5 seconds, all permutations are tried in
56 days assuming a 24-hour working day.
Combinations means the different ways we can choose a number of different objects from a group of objects where no order is involved, just the number of ways of choosing them.
In how many ways can we combine 2 flavors of ice cream if we can choose from strawberry (SJ,
vanilla (V), and chocolate (CJ without repeated flavors?We can combine them in 3 ways: SV,SC, VC. VS and SVare a combination of the same ice creams. The order doesn't matter. Vanilla on the top is the same as vanilla on the bottom.
The number of ways we can select r objects from a group of n objects is called a combination of n objects taken rat a time and is defined as n! / r!(n-r)!
The numberof ways we can select 3 people taken from agroup of 10 people is 120 (10!/3!(10-3J!).
The binomial distribution
Suppose we take a true-false test with 10 questions. We don't know anything about the subject. All we can do is guess. To pass this test, we must answer exactly 5 questions correct. Are we likely to do that by guessing?
How should we reason? Ask: How many equally likely independent outcomes are there when we guess? There are 2 possible outcomes. Either we are right or wrong. If the test
only had one question, the probability that we guess the right answer is 50%. The probability that we guess the wrong answer is also 50% {1- the probability of guessing the right answer).
What is the total number of equally likely outcomes? Since every question has 2 possible outcomes and there are 10 questions there are a total of 210 outcomes or 1,024 possible true false combinations. We can answer the test in 1,024 different ways. What is the number of favorable outcomes? There is only one way we can answer all 10 questions right (or wrong). They all have to be right (or wrong). The chance of getting all 10 answers right or wrong by guessing is therefore 1 in 1024. This means that if we took the test 1,024 times and guessed
283
randomly at the answers each time, just once in 1,024 times should we expect to get all 10
answers right or wrong.
In how many ways can we be right on 5 questions? Let's go back to combinations and ask: in how many ways can we select 5 questions if we can choose from 10 questions? There are 252 ways (10!/5!(10-5)!) we can answer 10 questions to get exactly a score of 5. Since each guess has a 50% probability of being right and there are 10 questions and we want to be right on exactly 5 of them and there are 252 equally likely ways we can answer 5 questions, then the probability that we answer exactly 5 questions right is (0.5)5 x (0.5)5 x 252 = 24.6%
Seeking Wisdom Page 39
