Addition and Multiplication of Probabilities
These are my notes on the addition and multiplication of probabilities.
Addition Rule
The addition rule is a tool for finding P(A or B), which is the probability that either event A occurs or event B occurs as the single outcome of a procedure. The word “or” in the addition rule is associated with the addition of probabilities.
Multiplication Rule
This section also presents the basic multiplication rule used for finding P(A and B), which is the probability that event A occurs and event B occurs. The word “and” in the multiplication rule is associated with the multiplication of probabilities.
Compound Event
A compound event is any event combining two or more simple events.
Addition Rule
Here is the notation for the addition rule. P(A or B) = P(in a single trial, event A occurs or event B occurs or they both occur).
Intuitive Addition Rule
To find P(A or B), add the number of ways event A can occur and the number of ways event B can occur, but add in such a way that every outcome is counted only once. P(A or B) is equal to that sum, divided by the total number of outcomes in the sample space.
Formal Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
Where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure.
Disjoint Events and the Addition Rule
Events A and B are disjoint or mutually exclusive if they cannot occur at the same time. That is, disjoint events do not overlap.
Event A - Randomly selecting someone for a clinical trial who is a male.
Event B - Randomly selecting someone for a clinical trial who is a female.
Disjoint Events
Event A - Randomly selecting someone taking a statistics course.
Event B - Randomly selecting someone who is a female.
Complementary Events and the Addition Rule
We use \(\bar{A}\) to indicate that event A does not occur. Common sense dictates this principle. We are certain with probability of 1 that either an event A occurs or does not occur, so it follows that |(P(A or \bar{A}) = 1. Because events \(A \text{and} \bar{A}\) must be disjoint, we can use the addition rule to express this principle as follows:
\[P(A or \bar{A}) = P(A) + P(\bar{A}) = 1 \]
Rule of Complementary Events
\[ P(A) + P(\bar{A}) = 1 \]
\[ P(\bar{A}) = 1 - P(A) \]
\[ P(A) = 1 - P(\bar{A}) \]
Multiplication Rule
P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)
P(B | A) represents the probability of event B occurring after it is assumed that event A has already occurred.
Multiplication Rule
P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)
P(B | A) represents the probability of event B occurring after it is assumed that event A has already occurred.
Intuitive Multiplication Rule
To find the probability that event A occurs in one trial and event B occurs in another trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B is found by assuming that event A has already occurred.
Formal Multiplication Rule
P(A and B) = P(B | A)
Independence and the Multiplication Rule
Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. Several events are independent if the occurrence of any does not affect the probabilities of the occurrence of the others. I A and B are not independent, they are said to be dependent.
Sampling
In the world of statistics, sampling methods are critically important.
Sampling with replacement: Selections are independent events.
Sampling without replacement: Selections are dependent events.
Treating Dependent Events as Independent
When sampling without replacement and the sample size is no more than 5% of the size of the population, treat the selections as being independent, even though they are actually dependent.
Redundancy
The principle of redundancy is used to increase the reliability of many systems. Our eyes have passive redundancy in the sense that if one of them fails, we continue to see. An important finding of modern biology is that genes in an organism can often work in place of each other. Engineers often design redundant components so that the whole system will not fail because of the failure of a single component
When randomly selecting an adult, A denotes the event of selecting someone with blue eyes. What do \(P(A)\) and \(P(\bar{A})\) represent?
\(.P(A)\) represents the probability of selecting an adult with blue eyes.
\(P(\bar{A}) represents the probability of selecting an adult who does not have blue eyes.
There are 15,958,866 adults in a region. If a polling organization randomly selects 1235 adults without replacement, are the selections independent or dependent? If the selections are dependent, can they be treated as independent for the purposes of calculations?
The selections are dependent because the selection is done without replacement.
Yes, because the sample size is less than 5% of the population.
When randomly selecting an adult, let B represent the event of randomly selecting someone with type B blood. Write a sentence describing what the rule of complements below is telling us.
\(P(B or \bar{B}) = 1\)
It is certain that the selected adult has type B blood or does not have type B blood.
A research center poll showed that 76% of people believe that it is morally wrong to not report all income on tax returns. What is the probability that someone does not have this belief?
.24
Find the indicated complement.
A certain group of women has a 0.2% rate of red/green color blindness. If a woman is randomly selected, what is the probability that she does not have this color blindness?
.9998
Use the data in the following table, which lists drive-thru order accuracy at popular fast food chains. Assume that orders are randomly selected from those included in the table.
A B C D
316 266 250 125
32 56 37 20
If one order is selected, find the probability of getting food that is not from restaurant A.
Add up all of B,C, and D then divide by all of A,B,C, D.
754/1098=.68
Use the data in the following table which lists drive-thru order accuracy at popular fast food chains. Assume that orders are randomly selected from those included in the table.
If one order is selected, find the probability of getting an order that is not accurate.
Add up incorrect orders and then total orders
A B C D
320 260 236 149
39 59 32 12
142/1107= .128
Use the data in the following table, which lists drive-thru order accuracy at popular fast food chains.
A B C D
321 280 244 129
39 51 30 14
If one order is selected, find the probability of getting an order from restaurant A or an order that is accurate. Are the events of selecting an order from restaurant A and selecting an accurate order disjoint events?
The formal addition rule is \( P(A or B) = P(A) + P(B) - P(A and B) \)
Accurate orders =974
Inaccurate orders from restaurant A=39
Add together to get 1013
1013/1108=.914
Use the data in the following table, which lists drive-thru order accuracy at popular fast food chains.
A B C D
367 255 206 176
45 53 22 28
If two orders are selected, find the probability that they are both from restaurant D
Assume that the selections are made without replacement, are the events independent?
\[ P(A and B) = P(A) * P(B | A) \]
Calculate total orders from all restaurants
Calculate orders from restaurant D
Divide orders from restaurant D by the total number of orders. This gives \(P(A)\)
- Assume that the selections are made with replacement
The events are independent and probability of event B stays the same regardless of event A
So, \( P(A and B) = \frac{204}{1152} * \frac{204}{1152} = .0314 \)
- Assume that the selections are made without replacement.
The probability of event A will be the same \(\frac{204}{1152}\)
When replacements are not used, the events are not independent and the probability of event B changes depending on the outcome of event A.
Since event A was selecting an order from D, the selected order does not get replaced, the number of orders from D and the total number of orders to choose from each side each decrease by 1 when choosing event B.
So:
\[ P(A) = \frac{204}{1152} \text{and} P(B | A) = \frac{204-1}{1152-1} \]
Multiply the probability of event A by event B
\[ P(A and B) = .0312 \]
Use the data in the following table, which lists drive-thru order accuracy at popular fast food chains.
A B C D
323 267 241 128
30 55 34 12
If two orders are selected, find the probability that they are both accurate.
- Assume that the selections are made with replacement. Are the events independent?
Calculate total number of orders: 1090
Accurate orders: 959
\[\frac{959}{1090} * \frac{959}{1090} = .7741 \]
- Assume that the selections are made without replacement. Are the events independent?
Because the selections are made without replacement, the events are dependent events.
The probability of each order being accurate is affected by the other orders.
The probability \(P(A)\} remains the same as in part A.
The probability \(P(B|A)\) must be adjusted to reflect that the first order was accurate and is not available for the second order.
Recall that originally there were 1004 accurate orders out of 1152.
After the first accurate order is selected, there are 1151 orders remaining of which 1003 are accurate.
\[ P(A and B) = \frac{959}{1090} * \frac{958}{1089} = .7740 \]
The events are not independent because the sampling is done without replacement
Use the data in the following table.
A B C D
321 260 243 121
35 52 32 14
If three orders are selected, find the probability that they are all from B.
\[(312 / 1078) * 3 = .0242 \]
Use the following results from a test for marijuana use, which is provided by a certain drug testing company. Among 145 subjects with positive test results, there are 29 false positive results. Among 157 negative results, there are 3 false negative results.
- How many subjects were in the study?
No Yes
pos= 29 145
neg= 157 3
How many subjects were included in the study?
Add the subjects who tested positive to those who tested negative= 302
How many subjects did not use marijuana?=183
What is the probability that a randomly selected subject did not use marijuana?183/302=.606
Among 132 subjects with positive test results, there are 32 false positive results
Among 168 negative results, there are 8 false negative results.
If one of the test subjects is randomly selected, find the probability that the subject tested negative or did not use marijuana.
32 100
160 8
Total subjects=300
Next, find the probability that a randomly selected subject tested negative
168/300
Now, find the number of subjects that did not use marijuana
Two groups did not use marijuana. True negatives and the false positives
160+32=192
Next, find the probability that a randomly selected test subject did not use marijuana.
Did not use=192/300
Next, find the probability that a randomly selected test subject tested negative and did not use it
160/300
Finally, use the formal addition rule to find the probability that a randomly selected subject tested negative or did not use it, rounding to 3 decimal places
168/300+192/300-160/300 = .667
The principle of redundancy is used when system reliability is improved through redundant components. Assume that a student’s alarm has a 16.0% daily failure rate.
- What is the probability that the student’s alarm clock will not work on the morning of an important exam?
To convert a percentage to a decimal number, remove the % symboland divide by 100.
For the stated failure rate of 16% remove the percent symbol and divide by 100.
16/100 = .160
So, the probability that the student’s alarm clock will not work on the morning of an important exam is .160.
- If the student has two such alarm clocks, what is the probability that they both fail on the morning of an important exam?
Use the formal rule of multiplication that states if P(A) is the probability of event A occurring and P(B|A) is the probability of B occurring given that A has occurred, the probability of both A and B occurring is given by:
\[P(A and B)=P(\bar{A})*P(\bar{A}|\bar{B}\]
The functioning of the second alarm clock is not affected by the failure of the first, so by definition they are independent events.
Multiply A and B together.
.160*.160=.0256
- What is the probability of not being awakened if the student uses three independent alarm clocks?
A * B * C = .160*.160*.160= .00410
- Do the second and third alarm clocks result in greatly improved reliability?
Compare the probability of one alarm clock not working to the probabilities of 2 or 3 alarm clocks not working. In general, when an event will occur with probability 1, it is called certain. An event occurring with probability less than or equal to .05 is called unlikely. An event occurring with probability 0 is called impossible.
Surge protectors p and q are used to protect a television. If there is a surge in the voltage, the surge protector reduces it to a safe level. Assume that each surge protector has a .88 probability of working correctly when a voltage surge occurs.
- If the two surge protectors are arranged in a series, what is the probability that a voltage surge will not damage the television?
With two independent surge protectors in series, the television will be protected unless both surge protectors fail. In other words, only one surge protector needs to work. Find the probability that only one surge protector works by calculating 1-P(p and q). This probability can be found by applying the multiplication rule for independent events.
\[P(A and B)=P(A)*P(B)\]
The probability that a surge protector works correctly is .88. The probability that a surge protector fails is calculated below.
1-.88=.12
The probability that one surge protector fails is .12. The probability that both surge protectors fail is the product of the probabilities that either one fails.
.12*.12=.0144
There is a .0144 probability that both surge protectors fail. The probability that the television is protected in a series configuration is the complement of the probability that both fail.
1-.0144=.9856
- If the two surge protectors are arranged in parallel, what is the probability that a voltage surge will not damage the television?
With two independent surge protectors in parallel, the television will be protected as long as both surge protectors work. The probability that the two independent surge protectors both work is found by applying the multiplication rule for independent events.
\[P(A and B)=P(A)*P(B)\]
The probability that a surge protector works correctly is .88. The probability that both surge protectors work is the product of the probabilities that both work correctly.
.88*.88=.7744
- Which arrangement should be used for better protection?
Series