HSS-CP.A.2STUDENT NOTES & PRACTICE WS #1–geometrycommoncore.com1
Independence
When working with probabilities we often perform more than one event in a sequence this is called a compound probability. Compound probabilities are more complex than a single event probability to compute because the first event might affect the probability of the second event happening.
For example, the probability of getting a head on a single flip of a coin is ½. If you flip the coin and get a head, the second flip’s probability of getting a head is still ½ because the results of the first flipdoes not in any way affect the second flip. The second flip has the exact same probability as if it was the first flip, ½. When the first action does not affect the second action’s probability in any way the events are known to be INDEPENDENT.
In contrast, if you have a jar of cookies with 7 chocolate chip cookies and 3 peanut butter cookies the probability of getting a chocolate chip is 7/10 and the probability of getting a peanut butter cookie is 3/10. Is the probability of getting a peanut butter cookie still 3/10, if you first pick out a chocolate chip cookie and eat it? Of course not, the probability of getting a peanut butter cookie now is 3/9 because a chocolate chip cookie is gone from the jar. The second selection is affected by the first selection, thus these two events are NOT INDEPENDENT.
Definition: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring (or vice versa).
Some other examples of independent events are:
-- Getting a head after tossing a coin AND selecting a purple marble from a bag.
-- Getting on head after tossing a coin AND rolling a 2 on a single 20-sided die.
-- Choosing a jack from a deck of cards, replacing it, AND then choosing aking as the second card.
These two events are independent - the die could roll any number of times and it would in no way influence the flip of the coin. Let us use a set list, a tree and a Venn diagram to understand this problem.
Set U = {1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T}
Set S = Rolling a 6 = {6H, 6T}
Set H = Heads = {1H, 2H, 3H, 4H, 5H, 6H}
Set SSet H = {6H}
P(6ANDH) = 1 / 12
Remember that AND represents the intersection.
/
To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, and then multiply the probabilities. This multiplication rule is defined symbolically below. Note that multiplication is represented by AND.
When two events, A and B, are independent, the probability of both occurring is:
P(A and B) = P(A) · P(B)
Example #1 / Example #2
What is the probability of rolling a 6 and
then rolling a 5?
P(S) = P(F) =
P(SandF) = P(S) P(F) = / Given a bag of marbles with 3 red, 2 green and 5 yellow. What is the probability of choosing a red, replacing it, and then choosing a green?
P(R) = P(G) =
P(R and G) = P(R) P(G) =
Example #3 / Example #4
What is the probability of getting a head on a coin flip and then choosing a purple marble from a bag that has 2 purple, 1 green, and 2 orange marbles?
These two events are independent. Let us use a tree diagram to understand and solve this problem. What is the P (Head and Purple)? / A true and false question is followed multiple choice question with possible four answers (1 correct & 3 wrong). What is the probability of getting both questions correct,P(CC)?
Understanding independence is critical to probability because we must always take into account how one event affects the next event.
NYTS (Now You Try Some)
1. Determine if the following are independent or not.
a) Event #1Event #2
The Date The Amount of Sunlight / b) Event #1Event #2
Income Education Level / c) Event #1Event #2
Your Height Your Favorite Color
(I)ndependent or (N)ot Independent / (I)ndependent or (N)ot Independent / (I)ndependent or (N)ot Independent
2. Determine the P (A and B) given that Event A and Event B are independent.
a) P(A) = 0.1, P(B) = 0.3 / b) P(A) = 0.25, P(B) = 0.8 / c) P(A) = 0.5, P(B) = 0.5
P(A and B) = ______/ P(A and B) = ______/ P(A and B) = ______
Mutually Exclusive and Independence
A common misunderstanding is that independence is the same thing as being mutually exclusive. I get why this is confusing, to be independent in a typical English language context means to be alone or separate which is basically what we understandmutually exclusive to mean. This definition of independence is NOT the mathematical one. Independence is about whether oneevent affects another event’s probability or not. Mutually exclusive sets are those that don’t share any elements and independentsetsare those that don’t impact each other’s probabilities.
Mutually exclusive is about the sharing of elements,
and independence, is about affecting each other.
But Events A and Event B are INDEPENDENT because
P(A) P(B) = (0.3)(0.5) = 0.15P(A AND B) = 0.15
So here is one example where mutually exclusive and independence obviously are two different things!! /
Replacement and No Replacement
The terms replacement and no replacement get used a lot in compound probabilities problems because they describe what you did with the first thing that you selected… did you put it back or did you keep it?
P (Getting a green marble, replacing it, and getting a green marble)Independent
P (Picking a black queen, not replacing it, and getting an ace)Not Independent
These two words are HUGE clues as to whether the events are going to be independent or not.
REPLACEMENTBecause the item is replaced, it resets the event back to the original arrangement and
no probabilities are altered. Thus REPLACEMENT tells us that the events are INDEPENDENT.
NO REPLACEMENTBecause the item is NOT replaced, the probabilities are altered. Thus NO
REPLACEMENT tells us that the events are NOT INDEPENDENT.
3. Determine if the following are independent or not.
a) picking a marble from a bag, replacing the marble and then picking again. / b) picking a marble from a bag and then spinning a 4 color spinner. / c) picking a card from a standard deck, not replacing the card and then picking again.
(I)ndependent or (N)ot Independent / (I)ndependent or (N)ot Independent / (I)ndependent or (N)ot Independent
Testing for Independence
We can use the formal relationship of P (A and B) = P(A) P(B) to test independence. Here are three examples of how we could use probabilities to determine if they are independent of each other.
Example #1P(A) = 0.8P(B) = 0.4P(A and B) = 0.2
These are Not Independent because P(A) P(B) = (0.8)(0.4) = 0.32 and this is not
the same as P(A and B) = 0.2 provided.
Example #2P(A) = 0.6P(B) = 0.5P(A and B) = 0.3
These are Independent because P(A) P(B) = (0.6)(0.5) = 0.30 and this is
the same as P(A and B) = 0.3 provided.
The same kind of test can occur but the information can be given in Venn diagram form.
Example #3
P(A) = 0.5
P(B) = 0.5
P(A AND B) = 0.25 / / Check to see if P(A) P(B) = P (A and B)
P(A) P(B) = (0.5)(0.5) = 0.25
P (A and B) = 0.25
INDEPENDENT
Example #4
P(A) = 0.7
P(B) = 0.2
P(A AND B) = 0.14 / / Check to see if P(A) P(B) = P (A and B)
P(A) P(B) = (0.7)(0.2) = 0.14
P (A and B) = 0.14
INDEPENDENT
Example #5
P(A) = 0.50
P(B) = 0.28
P(A AND B) = 0.20 / / Check to see if P(A) P(B) = P (A and B)
P(A) P(B) = (0.5)(0.28) = 0.14
P (A and B) = 0.2
NOT INDEPENDENT
NYTS (Now You Try Some)
4. Determine if the following are independent or not.
a) Independent or Not Independent / b) Independent or Not Independent / c) Independent or Not Independent
/ / P(A) = 0.45
P(B) = 0.6
P (A and B) = 0.27
5. Determine the missing probability given that the events are independent.
a) P(A) = 0.2 P(B) = 0.5
P (A and B) = ______ / b) P(A) = 0.4 P(A and B) = 0.25
P (B) = ______ / c) P(A and not B) = 0.2 P(A) = 0.3
P (A and B) = ______