Chpt 4 - Probability Flashcards
What is probability?
The measure of likeliness that a certain result of a chance experiment occurs
What is a chance experiment?
A process producing outcomes that vary randomly when repeated
What is the measure of likeliness that a certain result of a chance experiment occurs?
Probability
What is a process producing outcomes that vary randomly when repeated?
Chance experiment
What is an elementary outcome of a chance experiment called?
The sample point
What are the sample points of flipping a coin?
Head and tail
What is the collection of all sample points of a chance experiment called?
A sample space
What is the sample space denoted by?
capital S
What is the sample space when flipping a coin?
{head, tail}
What is the sample space when flipping 2 coins?
{HH, HT, TH, TT}
What is an event?
A specified result that may or may not occur
Such as flipping a coin and getting heads
How can an event be written out?
Give an example of flipping 2 coins and getting at least one head
A collection of outcomes
A = {HH, HT, TH}
Where A is the event, such as flipping a head, and the data within the brackets are possible outcomes in that event
What is the collection of outcomes for rolling higher than a 4 on a six sided dice?
A = {5, 6}
What is the probability of an event? Why is this difficult?
It’s the relative frequency of the event in a population
Remember it’s next to impossible to obtain the population data, so instead we use a sample
What is the range for a probability of an event
0 to 1
0 = an event that NEVER occurs
1 = an event that ALWAYS occurs
What is the f/N rule and why do we use it?
Assumes that the sample space has N sample points and these sample points are equally likely.
We use it because getting the entire population information needed for a relative frequency is difficult if not impossible to obtain, so as long as the sample is representative of the population we use a sample space
How do we denote a probability of event D?
P(D)
For event F, what is the probability of tossing 2 quarters and having the second toss be the same?
S = {HH, HT, TH, TT}
A = {HH, TT}
P(A) = 2/4 -> 1/2 or 0.5
What can we apply to events to produce new events?
Set operations
What is the complement set operation and what denotes it?
The complement of A is a set that includes sample points that are NOT in A but in the sample space S
It is denoted as notA or Ac (with the c superscript)
What is the intersection set operation and what denotes it?
The intersection of events A and B is a set that includes sample points that belong to BOTH A and B.
It is denoted as A&B or A ⋂ B
What is the union set operation and what denotes it?
The union of A and B is a set that includes sample points that are in A or B
It is denoted as AorB or A ⋃ B
What set operation is anything in a sample space that is not included within event A?
What is it denoted as?
Complement
It is denoted as notA or Ac (with the c superscript)
What set operation is anything that is included in only the overlapping area of 2 events (like the middle part of a Venn diagram)
What is it denoted as?
Intersection
It is denoted as A&B or A ⋂ B
What set operation is anything that is included in the entire A and B circles of a Venn diagram?
What is it denoted as?
Union
It is denoted as AorB or A ⋃ B
You are rolling a fair die. Event A is any number on the upper face less than 5 and event B is any number on the upper face more than 3 but less than 6.
What is the sample space of tossing the die?
S = {1, 2, 3, 4, 5, 6}
You are rolling a fair die. Event A is any number on the upper face less than 5 and event B is any number on the upper face more than 3 but less than 6.
What is event A? What is event B?
A = {1, 2, 3, 4}
B = {4, 5}
You are rolling a fair die. Event A is any number on the upper face less than 5 and event B is any number on the upper face more than 3 but less than 6.
What is in the complement of event A?
notA = {5, 6}
You are rolling a fair die. Event A is any number on the upper face less than 5 and event B is any number on the upper face more than 3 but less than 6.
What is in the union of A and B?
A ⋃ B = {1, 2, 3, 4, 5}
This is AorB
You are rolling a fair die. Event A is any number on the upper face less than 5 and event B is any number on the upper face more than 3 but less than 6.
What is the intersection of A and B?
A ⋂ B = {4}
This is A&B
There are 4 rabbits and 4 goats on a farm and 4 are chosen at random to be sent to the zoo. Suppose you observe the number of rabbits.
What is the sample space?
S = {0, 1, 2, 3, 4}
(remember we are only counting rabbits…if there are 0 rabbits going, then there are 4 goats going; 3 rabbits would result in 1 goat going)
There are 4 rabbits and 4 goats on a farm and 4 are chosen at random to be sent to the zoo. Suppose you observe the number of rabbits.
Event A has at least half the chosen animals as rabbits; event B is at least half the goats chosen.
What are the 2 events?
A = {2, 3, 4}
B = {0, 1, 2,}
Remember we are still only looking at rabbits, so if there is 1 rabbit, then 3 goats are going
There are 4 rabbits and 4 goats on a farm and 4 are chosen at random to be sent to the zoo. Suppose you observe the number of rabbits.
Event A has at least half the chosen animals as rabbits; event B is at least half the goats chosen.
What is the union of these events?
AorB = {0, 1, 2, 3, 4}
Remember we are still only looking at rabbits, so if there is 1 rabbit, then 3 goats are going
This is A ⋃ B
There are 4 rabbits and 4 goats on a farm and 4 are chosen at random to be sent to the zoo. Suppose you observe the number of rabbits.
Event A has at least half the chosen animals as rabbits; event B is at least half the goats chosen.
What is the intersection of these events?
A&B = {2}
Remember we are still only looking at rabbits, so if there is 1 rabbit, then 3 goats are going
This is A ⋂ B
There are 4 rabbits and 4 goats on a farm and 4 are chosen at random to be sent to the zoo. Suppose you observe the number of rabbits.
Event A has at least half the chosen animals as rabbits; event B is at least half the goats chosen.
Determine the A&(notB)
notB = {3, 4}
A = {2, 3, 4}
So:
A&(notB) = {3, 4}
What is the equation for the complement rule and what does this help us find?
P(notA) = 1 - P(A)
The complement rule helps us find the probability of the complement of an event, and we can find it as long as we know the probability of an event
If event A occurs 40% of the time, what is the probability of the complement of A?
P(notA) = 1 - P(A)
=1 - 0.40
= 0.60
What helps us determine the probability of the complement of an event?
Complement rule
What rule is used to find the probability of the union of 2 events?
Addition rule
What is the equation for the addition rule and what does it help us find?
P(AorB) = P(A) + P(B) - P(A&B)
Helps us find the probability of the union of 2 events
For the addition rule:
P(AorB) = P(A) + P(B) - P(A&B)
Why do we subtract the P(A&B)?
If we don’t, we count the overlapped area twice, as it is part of both P(A) and P(B)
Event A occurs 25% of the time and event B occurs 20% of the time. Event A&B occurs 5% of the time.
What is the probability of the union of event A and B?
P(AorB) = P(A) + P(B) - P(A&B)
=0.25 + 0.20 - 0.05
=0.40
What does it mean if event A and event B have no common sample points?
Mutually exclusive
What are mutually exclusive events?
Events with no common sample points
What is the special addition rule for mutually exclusive events? Why does this differ from the general addition rule
P(AorB) = P(A) + P(B)
The special addition rule does not need to subtract the overlapping areas of A and B because it is used for events that are mutually exclusive, and therefore has no overlapping areas
There are 4 rabbits and 4 goats on a farm and 4 are chosen at random to be sent to the zoo. Suppose you observe the number of rabbits.
Event A has at least half the chosen animals as rabbits; event B is at least half the goats chosen.
Are events A and B mutually exclusive?
A = {2, 3, 4}
B = {0, 1, 2} (remember we are counting the rabbits that would be taken…so 1 rabbit taken would mean 3 goats are taken)
A&B = {2}
Therefore A and B are not mutually exclusive
There are 4 rabbits and 4 goats on a farm and 4 are chosen at random to be sent to the zoo. Suppose you observe the number of rabbits.
Event A has at least half the chosen animals as rabbits; event B is at least half the goats chosen.
Are events A and not(B) mutually exclusive?
A = {2, 3, 4}
notB = {3, 4}
A¬B = {3, 4}
Therefore events A and notB are not mutually exclusive
What is conditional probability and how is it denoted?
A given B is the probability that event A occurs if event B occurs
B is the condition in this equation.
It is denoted as P(A|B)
What is the difference between P(A) and P(A|B)?
P(A) is the probability that A occurs without any other information
P(A|B) is the probability that A occurs given the information of event B
What does P(A|B) mean?
Probability of A given B
Its a conditional probability that event A occurs if event B occurs
You’re tossing a fair die.
Event A is the number on the upper face is 6 and event B is the number on the upper face is odd.
What is P(A|B)?
A = {6}; P(A) = 1/6
B = {1, 3, 5}; P(B) = 3/6 = 1/2
A&B = 0
P(A|B) = P(A&B)/P(B)
= 0 / (1/2)
= 0
What is the conditional probability rule use for? What is the rule and what is the equation we use for it?
Given the probability of the intersection of events A & B and the probability of B, we can find the conditional probability of A given B
P(A|B) = P(A&B) / P(B)
Used to calculate conditional probabilities
What are the 3 rules that prove that events A and B are independent?
P(A|B) = P(A)
P(B|A) = P(B)
P(A&B) = P(A) x P(B)
If any of the above equations is true, the events are independent.
What is the general multiplication rule?
What is it an adaptation of?
P(A&B) = P(A|B) x P(B)
Adaptation of conditional probability rule:
P(A|B) = P(A&B)/P(B)
What does this rule mean?
P(A|B) = P(A)
What does it prove?
Means that the probability that A occurs is NOT affected by the occurrence of event B
It proves the events are independent
What does this rule mean?
P(B|A) = P(B)
What does it prove?
Means that the probability that B occurs is NOT affected by the occurrence of event A
It proves the events are independent
What does this rule mean?
P(A&B) = P(A) x P(B)
What does it prove?
P(A&B) = P(A) x P(B)
Derived from the multiplication rule:
P(A&B) = P(A|B) x P(B)
Put together they are:
P(A) x P(B) = P(A|B) x P(B)
or P(A) = P(A|B)
It if is true for the problem you’re working on, events A and B are independent
Consider you are throwing a fair die.
Event A is tossing a 2, 4, or 6 and event B is tossing a 5 or 6.
Are these events mutually exclusive? Are they independent?
S = {1, 2, 3, 4, 5, 6}
A = {2, 4, 6} P(A) = 1/2
B = {5, 6} P(B) = 1/3
Are they mutually exclusive?
A&B = {6} P(A&B) = 1/6
There is overlap between A and B so they are NOT mutually exclusive
Are they independent?
(conditional probability rule)
P(A|B) = P(A&B)/P(B)
=(1/6) / (1/3) = 1/2
P(A|B) = P(A)
1/2 = 1/2
Yes, they are independent, because the probability of A
What does it mean if we say 2 events are independent?
The occurrence of one event does NOT affect the chances of the occurrence of the other event
How do we find the joint probability if we know P(A) and P(A|B)?
P(A&B) = P(A) x P(A|B)
This is based on the conditional probability rule:
P(A|B) = P(A&B) / P(B)
It’s the same as the general multiplication rule:
P(A&B) = P(A|B) x P(B)
How do we find the joint probability for 2 known independent events?
P(A&B) = P(A) x P(B)
This is based on the conditional probability rule:
P(A|B) = P(A&B) / P(B)
It’s the same as the general multiplication rule:
P(A&B) = P(A|B) x P(B)
But because with independent events, P(A|B) = P(A), we can just sub it out
How do we find the joint probability if we know events A and B as well as the union of P(AorB)?
P(A&B) = P(A) +P(B) - P(AorB)
Its based on the general addition rule which is:
P(AorB) = P(A) +P(B) - P(A&B)
How can the probability and A and B be calculated if the population data are known and summarized in a contingency table?
For example, we want to know what the probability of being both a man and being single. In the table, we know:
Single Total Men 10 240 Women 12 260 Total 22 500
We can use just the basic numbers, rather than find the probability of each variable
For this example
P(Men&Single) = 10/500
What is joint probability?
It is the probability of the intersection of 2 events A and B
It’s basically finding the overlap area of the Venn Diagram of 2 circles
Based on the following table, what is the probability that a woman is single?
Single Total Men 10 240 Women 12 260 Total 22 500
Joint Probability
P(W&S) = 12/500
Based on the following table, what is the probability of being a woman?
Single Total Men 10 240 Women 12 260 Total 22 500
P(W) = 260/500
Based on the following table, what is the probability of being single given that the person is a man?
Single Total Men 10 240 Women 12 260 Total 22 500
P(S|M) = P(S&M) / P(M)
= 10/240
Based on the following table, what is the probability of being a man given that the person is single?
Single Total Men 10 240 Women 12 260 Total 22 500
P(M|S) = P(M&S) /P(S)
= 10/22
What is the special multiplication rule?
Used to determine the joint probability when the 2 events are independent and is:
P(A&B) = P(A) x P(B)
It’s an adaptation of the general multiplication rule:
P(A&B) = P(A) x P(A|B)
What can be used to help count the number of sample points in a sample space if there are hundreds/thousands of sample points without listing them all?
Counting rules :)
What are the 3 counting rules used in this class
Basic counting rule
Combination rule
Permutation rule
What is the basic counting rule?
If there are “m” possible outcomes for one thing, and there are “n” possible outcomes for another thing, then there are “mn” possible outcomes for both things
Suppose you had a license plate that consists of one letter (A-Z) and one number (0-9). Using the basic counting rule, how any possible license plate options?
m = 26 letters
n = 10 numbers
mn = 26 x 10 = 260
What is the general counting rule?
An expansion on the basic counting rule
For example:
1st stage = n1 outcomes
2nd stage = n2 outcomes
3rd stage = n3 outcomes
….
nth stage = nk outcomes
General counting rule:
n1 x n2 x n3 x … nk = total outcomes
What is a combination in the context of the combination rule
A combination is a selection of all or part of a set of objects WITHOUT REGARD TO THE ORDER in which they are selected. So order is not important. It’s not important, because the objects in question are all the same
An example would be you randomly select 10 paramedics to give them each $20 Tim Hortons gift cards.
What is the combination rule?
What is the equation for it?
The number of possible combinations of r objects from m objects
The equation is:
mCr = m!
———–
r!(m-r)!
The easy way to say this would be choose….so m choose r….or 10 choose 3 to help fill in the numbers
How many ways can I give 3 coffee mugs to 8 paramedics?
It’s a combination (or a choose), because the coffee mugs are all the same. So the equation is
mCr = m!
———–
r!(m-r)!
8C3 = 8!
———–
3!(8-3)!
8C3 = 56
What is a permutation in the context of the permutation rule
A selection of all or part of a set of objects WITH REGARD TO THE ORDER in which they are selected. So here the order is important, and that is because the objects are not the same.
For example, I have a Tims Card, a coffee mug, and a lb of ground coffee to give away to 10 randomly selected paramedics.
Another example is 3 new construction projects to be given to 10 companies and no company can be selected twice. The construction projects will all be different, so its a permutation
What is the permutation rule?
What is the equation for it?
The number of possible permutations (ordered arrangements) of r objects from m objects
The equation is:
mPr = m!
———–
(m-r)!
The easy way to say this would be pick….so m pick r….or 10 pick 3 to help fill in the numbers
There is a lottery with 50 tickets sold. There are 3 different prizes to be awarded to the first/second/third winners. How many possible outcomes are there?
Order matters because the prizes are different
mPr = m!
———–
(m-r)!
50P3 = 50!
——–
(50-3)!
50P3 = 117 600
I have a Tims Card, a coffee mug, and a lb of ground coffee to give away to randomly selected paramedics. How many way ways can I give an object to 8 randomly selected paramedics?
It’s a permutation (or a pick), because the objects are all different. So the equation is
mPr = m!
———–
(m-r)!
8P3 = 8!
———–
(8-3)!
8P3 = 336
What is the main difference between the combination and permutation rules?
Combinations are used when the order doesn’t matter as the objects are the same
Permutations are used when the order matters as the objects are different
A lottery with 50 tickets sold has 3 equivalent prizes to be awarded to the first/second/third prize winners. How many possible outcomes are there?
mCr = m!
———–
r!(m-r)!
50C3 = 50!
————
3!(50-3)!
50C3 = 19 600
A university has 20 departments and 2 students were selected from each department to compete for a student scholarship which will be awarded to 3 students.
How many possible ways are there to distribute the scholarship?
40C3 = 40!
————
3!(40-3)!
40C3 = 9880
A university has 20 departments and 2 students were selected from each department to compete for a student scholarship which will be awarded to 3 students.
How many possible ways are there if all the winners are from different departments?
Step 1: Determine how many ways the departments might be chosen
20C3 = 20!
————
3!(20-3)!
20C3 = 1140
Step 2: Determine the ways to select the students from each of the winning departments
2 x 2 x 2 = 8
Step 3: Basic counting rule to determine ways to distribute the scholarship
1140 x 8 = 9120
A university has 20 departments and 2 students were selected from each department to compete for a student scholarship.
If 5 winners are selected at random from all 40 students, what is the probability that no 2 winners are from the same departments?
Step 1: Determine outcomes for selecting 5 of 40 students
40C5 = 40! / (5!(50-5)!)
40C5 = 658 008 (this is N)
Step 2: Determine possible ways for Event A
(20C5) x (2x2x2x2x2)
15504 x 32 = 49 600 (this is f)
Step 3: f/N rule
49 600 / 658 008
=0.075379
