Week 2

Question 1

Answer 1

Question 2

d. Show that 𝐴 ∪ (𝐴̅ ∩ 𝐵) = (𝐴 ∪ 𝐵)

Answer 2

Question 3

Answer 3

Question 4

Answer 4

Question 5

Answer 5

Question 6

Answer 6

Question 7

Answer 7

The sum of the probabilities of collectively exhaustive events must be equal to 1.
The number of combinations of 𝑥 objects chosen from 𝑛 is equal to the number of
combinations of (𝑛 − 𝑥) objects chosen from 𝑛, where 1 ≤ 𝑥 ≤ 𝑛 − 1.

Question 8

Answer 8

Question 9

Answer 9

Question 10

Answer 10

Question 11

Answer 11

Question 12

Answer 12

Question 13

Answer 13

Question 14

Answer 14

Question 15

Answer 15

Question 16

Answer 16

Question 17

Answer 17

Question 18

Answer 18

Question 19

Answer 19

Question 20

Answer 20

Question 21

Answer 21

Question 22

Answer 22

This article discusses how randomised tests when applied to a low-incidence
population (like the general public without coronavirus symptoms) will lead to an
overstatement of the percentage of the population who is infected with coronavirus. This is
because the number of positives observed will be driven by false positives , unless the test has
a near perfect specificity. This is called the base-rate fallacy. This is not problematic when
testing a high -incidence population, such as those with coronav irus symptoms. The article
advocates separately reporting the number of positives to portray a more accurate picture.

Question 23

Answer 23

Question 24

Answer 24

Question 25

Answer 25

This is a famous problem based on an old television show called “Let’s Make a Deal. ” You might be fixated on the fact that there are now only two doors left, so many people conclude there must be a 50:50 chance of winning whether you play a strategy of switching or a strategy of staying. However , you should always switch. This is because the host’ s actions will allow you to update your beliefs about the location of the car .

Question 26

what is a random experiment

Answer 26

process leading to two or more possible outcomes without knowing exactly which outcome will occur

Question 27

what is the sample space

Answer 27

its the set of all basic outcomes (s)

Question 28

what is an event?

Answer 28

E any subset of basic outcomes from the sample space it occurs if the randome experiment results in one of its basic oucomes occur

Question 29

what is an intersection?

Answer 29

is the set of all basic outcomes in the sample space that belong to both A and B

Question 30

what does mutually exclusive mean?

Answer 30

if the events A and B have no common basic outcomes the events cant occur together

Question 31

what is a union?

Answer 31

the union of two events in the sample space (A and B) is the set of basic outcomes that belong to at least one of the events occurs if and only if either A or B occur

Question 32

what does collectively exhaustive mean?

Answer 32

if the union of several events covers the entire sample space (s)

Question 33

what is the complement of an event?

Answer 33

represents all outcomes in the sample space not included in A denoted by A bar

Question 34

what is probability?

Answer 34

quantifiies how likely an event is to occur ranges from 0 to 1

Question 35

classical probability

Answer 35

proportion of times that an event will occur assumes all outcomes in the sample space are equally likely to occur

Question 36

relative frequency probability

Answer 36

indicates how often an event will occur compared to other events

Question 37

subjective probability

Answer 37

reflects personal judgment on expertise when estimating the likelihood of an event - based on intuition, opinions or expert knowledge

Question 38

probability postulates (rules for probability)

Answer 38

1. if any A is any event in sample space must be between 0 and 1 2. P (A) is the sum of probability of basic outcomes 3. P (sample space) = 1

Question 39

what are the consequences of the postulates?

Answer 39

if A and B are mutually exclusive the probability of their union is the sum of their individual probabilities if A and B are collectively exhaustive, the probability of their union = 1 as their union is the whole sample space

Question 40

what are the solution steps for Baye's Theorem?

Answer 40

1. define the subset events from the problem 2. define the probabilities and condtional probabilities for the events derived in step 1 3. compute the complement of the probabilities 4. formally state and apply Bayes theorem to compute the solution probability

Question 41

what is Bayes theorem?

Answer 41

method for updating conditional probabilities based on new information

Question 42

what is the complement rule?

Answer 42

P (A union A bar) = P (A) + P(A bar) P (A bar) = 1 - P(A)

Question 43

what is the additon rule of probabilities?

Answer 43

P A union B = P(A) + P (B) - P (A ∩ 𝐵)

Question 44

what is conditional probability?

Question 45

what is the multiplication rule of probability?

Answer 45

Question 46

what is statisitcal independence?

Answer 46