Probability

Question 1

is the default for experiments with replacement or without replacement

Answer 1

without replacement

Question 2

probability of event x = ?

Answer 2

[number of outcomes in which x occurs]/[total number of outcomes in the experiment]

Question 3

probability properties:

Answer 3

1. probability ranges from [0,1] inclusive
2. probability=0 means the outcome cannot occur
3. probability =1 means the outcome must occur

Question 4

what is the probability of the whole sample space?

Answer 4

*screen capture
1, since it is the set of all possible outcomes of the experiment

Question 5

probability description:

Answer 5

the likelihood that an event or a series of events will occur

Question 6

experiment:

Answer 6

an act whose outcomes are uncertain
ex: picking marbles from a jar

Question 7

outcome:

Answer 7

the result of an experiment

Question 8

event:

Answer 8

an outcome or a set of outcomes

Question 9

complementary events

Answer 9

two events that share no outcomes but together cover every possible outcomes are said to be complementary.
ex: in a jar of 20 blue marbles and 60 red marbles, picking a blue marble is complementary to picking a red marble (since if the marble is blue, it cant be red and vice versa). Since the event and its complement cover the entire probability space, the probability of an event and its complement sum to 1.

Question 10

complementary notation

Answer 10

for two complementary events, we refer to the first event as A and the complementary events as A’. P(A)+P(A’) = 1 or P(A) + P(not A) =1

Question 11

two events are complementary if and only if…

Answer 11

they are the only two possibilities that can occur

Question 12

probablility of a AND b means what?

Answer 12

AND means multiply when we are dealing with independent events

Question 13

what does it mean for two events to be independent?

Answer 13

two events A and B are independent if the fact that event A occurs does not change the probability that event B will occur

Question 14

for independent events A, and B – P(A and B) = P(A)*P(B)

Answer 14

this holds for any n number of events

Question 15

mutually exclusive != independent

Answer 15

splitting an apple with an arrow on the first shot and not splitting an apple with an arrow on the first shot are mutually exclusive but NOT independent
splitting an apple with an arrow on the first shot and not splitting an apple with an arrow on the second shot are not mutually exclusive and are independent

Question 16

two events A and B are dependent if…

Answer 16

the occurrence of the first event affects the occurrence of the second event in such a way that the probability of the second event is changed

Question 17

formula for dependent events –

Answer 17

P(A and B) = P(A) * P(B|A),
where P(A and B) is the probability that event A and event B will occur, P(A) is probability that event A will occur, and P(B|A) is probability that event B will occur given that event A has already occurred

Question 18

what does mutually exclusive mean?

Answer 18

means two events cannot occur together at the same time
ex: cannot role a heads and tails on the same flip of a coin

Question 19

addition rule concerning mutually exclusive events

Answer 19

when events A and B are mutually exclusive the probability that event A happens OR event B happens can be determined with the formula:
P(A or B) = P(A) + P(B)
the formula can be extended to account for more than two mutually exclusive events as well

Question 20

addition rule concerning events that are NOT mutually exclusive

Answer 20

P(A or B) = P(A) + P(B) - P(A and B)
since they are not mutually exclusive and there is a chance that they could both occur at the same time, when trying to find the probability of just one or the other occurring we have to subtract out the probability that they both occur

Question 21

shortcut for determining the probability for something with multiple outcomes

Answer 21

probability = (# of outcomes producing the event)*(probability of one outcome)

ex: a fair coin is flipped 3 times, what is the probability that the coin lands on heads exactly twice?
Step 1: figure out the number of outcomes producing the event - it will be some arrangement of H, H, and T. How many ways are there to arrange those outcomes? Since there are three items, where one of the items is repeated twice, there are (3!)/(2!) = 3 ways for these outcomes to be arranged.
Step 2: probability of one outcome. this can be calculated by putting the items in any arrangement -> P(H)P(H)P(T) =.5.5.5 = 1/8
Step 3: Multiply 3*(1/8)= 3/8

Question 22

how do you handle “at least” probability problems

Answer 22

ex: a fair coin is tossed 3 times, what is the probability that it lands on heads at least two times?
break this out into the two possible scenarios: it can land on heads 2 times or 3 times, find the probability of each and then add because it is “or”

Question 23

handing the special case of “at least 1” occurrences

Answer 23

ex: a fair coin is tossed 5 times, what is the probability that it lands on heads at least one time?
in this case, instead of summing up the scenario of heads 1 time + heads 2 times +… + heads 5 times, we can notice that since it landing on heads 0 times is the complement of the coin landing on heads at least 1 time (1, 2, 3, 4, or 5 times), we can treat these as collectively exhaustive and mutually exclusive events. So, since the probability of two complements is 1, we can say that 1 = P(0 heads) + P(at least one head), and P(at least one head) = 1 - P(0 heads), which is easy to compute.

Question 24

faster way to deal with dependent probability problems with multiple outcomes –>

Answer 24

consider using the combination method
probability of event = (number of favorable outcomes)/(total number of outcomes)

Question 25

how to do probability questions when some number of items MUST be selected?

Answer 25

in the numerator, when coming up with the total number of ways that outcome could occur, assume it has occurred and set up the combination or permutation that way, i.e. assume you have chosen them in the subgroup, so your combination problem will have you choosing less from a smaller pool.
ex: in a group of flowers there are 3 tulips, 3 roses, and 2 daisies. If a bouquet will be created by randomly selecting 5 of these flowers, what is the probability that both daises will be in the bouquet?
probability = (# favorable outcomes)/(total outcomes)=(# ways that some number of items MUST be selected)/(# of ways that all items can be selected)
for the numerator we want to figure out how many ways the 5 flowers can be chosen, assuming that two of them are daises, so this is “6 choose 3”. the denominator is straightforward - “8 choose 5)

Question 26

review about how to set up combination equation when a certain set of items MUST not be selected

Answer 26

mentally eliminate the items (those that will not be selected) from the main pool, this does not affect the number of items being chosen into the subgroup though

Question 27

formula for probability that some number of items will NOT be selected

Answer 27

probability = (total number of ways to make the selection when some number of items will not be selected)/(total number of ways that all items can be selected)

Question 28

how do you create the numerator of probability question when some number of items in a set must be chosen while another number of items CANNOT be chosen

Answer 28

screen capture

Question 29

quadratic equations may come up in probability because you may be given the probability of some sequence happening and be asked to identify how many items there were to start with.. see example screenshot

Answer 29

screenshot

Question 30

what is another way to think about probability formula?

Answer 30

probability is a fraction where numerator is number that expresses the number of favorable outcomes and denominator is number that expresses the total number of outcomes

Question 31

is probability a ratio?

Answer 31

yes it is the number of favorable outcomes to the total number of outcomes..
ex: if the probability of removing a blue candy from a jar of candies is 1/4, we know that there are x occurrences of the blue candy for every 4x occurrences of candies in the jar, and that for every 3 non-blue candies in the jar there are 4 candies

*see screenshot for an example of this coming into play

Question 32

see screenshot

Answer 32

see screenshot

Question 33

for probability questions involving permutation required to solve them, don’t forget the circular permutation formula!

Answer 33

there are (k-1)! ways to arrange k items in a circle, vs k! if they were to be arranged in a line..

Question 34

probability of creating codes

Answer 34

recall from the permutations section that the if you were to create a 7 letter code from all the letters of the alphabet (with no restrictions on repeated letters, etc), that there would be 26 choices for the first letter, 26 for the second, etc, so there are 26^7 ways to create the 7 letter code. what then is the probability of choosing the particular code VPNTXDW? p = # of favorable outcomes/ # of possible 7 letter arrangements = 1/(26^7)

Question 35

what is the probability of choosing the following pair of seven letter codes formed from letters of the alphabet (with no restrictions on repeating letters): VPNTXDW or JKLMFGW

Answer 35

probability of the first is 1/(26^7) and probability of the second is also 1/(26^7), so probability of either is 2/(26^7)

Question 36

what is the translation of overlapping sets to probability? for example, what is the formula to be used in the following problem: out of 50 students in a class, 20 take AP history, and 32 take AP biology. If 12 students take neither class, how many students take both?

Answer 36

use the formula for when A and B make up 2 sets - Total # of unique elements = # of A + # of B - # of both + # of neither. So, 50 = 20 + 32 - both +12 -> both = 14. This translate to probability in the following way:
1 = P(A) + P(B) - P(A and B) + P(neither A nor B), this is because these four events cover the entire probability space associated with A and B and thus sum to 1.

Question 37

what is the alternative formula for total number of elements?

Answer 37

total # = # of A only + # B only + # both + # of neither , how is this translated to probability for 2 events A and B?
1 = P(A but not B) + P(B but not A) + P(A and B) + P(neither A nor B)

Question 38

when events are not mutually exclusive, it is important to remember which probability formula?

Answer 38

P(A or B) = P(A) + P(B) - P(A and B)

Question 40

Minimizing and maximizing probability

Answer 40

See screenshot