statistics - topic 2: probabiltity

Question 1

what is a random experiment?

Answer 1

a process leading to an uncertain outcome

Question 2

what is a basic outcome?

Answer 2

a possible outcome of a random experiment

Question 3

what is a sample space?

Answer 3

the collection of all possible outcomes of a random experiment

Question 4

what is an event?

Answer 4

any subset of basic outcomes from the sample space

Question 5

what is an intersection of events?

Answer 5

the intersection is the set of all outcomes that belong to both events in a sample space

Question 6

what is mutually exclusive events?

Answer 6

two events are mutually exclusive if they have no basic outcome in common ie the two events cannot happen simultaneously

Question 7

what is the union of events?

Answer 7

If A and B are two events in a sample space , then the union of A and B , is the set of all outcomes in that belong to either A or B ie the everything in both events including the intersection

Question 8

what is collectively exhaustive events?

Answer 8

the events completely cover the sample space ie atleast one of the events must occur

Question 9

what is a complement of an event A ?

Answer 9

the complement of an event A is the set of all basic outcomes in the sample space that do not belong to A. it is denoted by A with a dash over it

Question 10

what are the three approaches of assessing the probability of an uncertain event?

Answer 10

1- classical probability
2- relative frequency probability
3- subjective probabilty

Question 11

what is classical probability?

Answer 11

it assumes all outcomes in the sample space are equally likely to occur

Question 12

what is the formula for the classical probability?

Answer 12

𝑃(𝐴)=𝑁_𝐴/𝑁
=(𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑡ℎ𝑎𝑡 𝑠𝑎𝑡𝑖𝑠𝑓𝑦 𝑒𝑣𝑒𝑛𝑡 𝐴) / (𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒)

Question 13

what are the rquirmenet sto work out the classical probabiltiy of event A?

Answer 13

Requires a count of the outcomes that satisfy event 𝐴 and the total number of outcomes in the sample space

Question 14

what is total number of possible orderings of n objects?

Answer 14

n! = n(n-1)(n-2)…..(2)(1)

Question 15

what is the permutations?

Answer 15

the number of possible arrangements when 𝑘 objects are to be selected from a total of 𝑛 objects and arranged in order [with (𝑛–𝑘) objects left over]

Question 16

what is the formula for the number of permutations?

Answer 16

𝑃_𝑘^𝑛=𝑛(𝑛−1)(𝑛−2)…(𝑛−𝑘+1)=𝑛!/(𝑛−𝑘)!

Question 17

what is the combinations?

Answer 17

the number of combinations of 𝑘 objects chosen from 𝑛 is the number of possible selections that can be made

Question 18

what is the formula of combinations?

Answer 18

𝐶_𝑘^𝑛= (𝑃_𝑘^𝑛)/𝑘!
=𝑛! / [ 𝑘!(𝑛−𝑘)! ]

Question 19

what is the difference between combinations and permutations?

Answer 19

the order of the events matter in permuations whereas it does not matter for combinations. for example permutations will class AB and BA as seperate whereas the combinations will class them as the same

Question 20

what is the relative frequency probabiltity?

Answer 20

The limit of the proportion of times that an event 𝐴 occurs in a large number of trials, 𝑛

Question 21

what is the formula for relative frequency probability?

Answer 21

𝑃(𝐴)=
〖𝑙𝑖𝑚〗(𝑛→∞) 𝑛_𝐴/𝑛=〖𝑙𝑖𝑚〗(𝑛→∞) (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑡ℎ𝑎𝑡 𝑠𝑎𝑡𝑖𝑠𝑓𝑦 𝑒𝑣𝑒𝑛𝑡 𝐴)/(𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠 𝑜𝑟 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠)

Question 22

what is the subjective probability?

Answer 22

An individual opinion or belief about the probability of occurrence

Question 23

what is the complement rule?

Answer 23

𝑃(𝐴̅ )=1− 𝑃(𝐴) i.e. 𝑃(𝐴) + 𝑃(𝐴̅ ) =1

Question 24

what is the addition rule?

Answer 24

The probability of the union of two events, 𝐴 and 𝐵, is:
𝑃(𝐴∪𝐵)=𝑃(𝐴)+𝑃(𝐵)−𝑃(𝐴∩𝐵)

Question 25

what is conditional probability?

Answer 25

A conditional probability is the probability of one event, given that another event has occurred

Question 26

what is the formula for a probility of A given B has occured?

Answer 26

𝑃(𝐴|𝐵) = 𝑃(𝐴∩𝐵 )/ 𝑃(𝐵)

Question 27

what is the formula for the probability of B given A?

Answer 27

𝑃(𝐵|𝐴)=𝑃(𝐴∩𝐵)/𝑃(𝐴)

Question 28

what is statistically independent?

Answer 28

Two events are statistically independent if and only if the probability of one event is not affected by the other event

Question 29

what is the formula for when two events are statistically independent?

Answer 29

𝑃(𝐴∩𝐵)=𝑃(𝐴)𝑃(𝐵)

Question 30

if A and B are independent then what is p(A) and P(B) equal to given they are greater then 0?

Answer 30

If A and B are independent, then
𝑃(𝐴|𝐵)=𝑃(𝐴) if 𝑃(𝐵)>0
𝑃(𝐵|𝐴)=𝑃(𝐵) if 𝑃(𝐴)>0

Question 31

what are joint probabilities?

Answer 31

they are the intersection probabilities

Question 32

what are marginal probabilties?

Answer 32

Marginal probabilities are the probabilities for individual events:
𝑃(𝐴_𝑖)=𝑃(𝐴_𝑖∩𝐵_1)+𝑃(𝐴_𝑖∩B2) + … + 𝑃(𝐴_𝑖∩𝐵_𝐾)
Where 𝐵_1,𝐵_2,…,𝐵_𝐾 are 𝐾 mutually exclusive and collectively exhaustive events

Question 33

what is the multiplication rule?

Answer 33

𝑃(𝐴_𝑖∩𝐵_𝑖 )=𝑃(𝐴_𝑖 |𝐵_𝑖)𝑃(𝐵_𝑖 )

Question 34

what is the bayes theoren

Answer 34

𝑃(𝐵_𝑖 |𝐴_𝑖 )=𝑃(𝐴_𝑖 |𝐵_𝑖 )𝑃(𝐵_𝑖 )/𝑃(𝐴_𝑖 )

Question 35

how can the bayes thereom be written using joint probabilties?

Answer 35

𝑃(𝐵_𝑖 |𝐴_𝑖 )=[𝑃(𝐴_𝑖 |𝐵_𝑖 )𝑃(𝐵_𝑖 ) ] / [(𝑃(𝐴_𝑖 |𝐵_1 )𝑃(𝐵_1 )+𝑃(𝐴_𝑖 |𝐵_2 )𝑃(𝐵_2 )+…+𝑃(𝐴_𝑖 |𝐵_𝐾 )𝑃(𝐵_𝐾 ) )]