chapters 5 and 6 probability and binomial distribution

Question 1

what is a experiment

Answer 1

• repeatable process that gives rise to a number of outcomes

Question 2

what is an event

Answer 2

• a set/collection of one or more of these outcomes

Question 3

what is a sample space

Answer 3

set of all possible outcomes

Question 4

what does it mean if events are “equally likely”

Answer 4

• the probability of an event is the number of outcomes in event divided by total number of possible outcomes

Question 5

what is the purpose of using a venn diagram

Answer 5

• used as a visual representations of events happening

Question 6

what does the rectangle labelled S signify

Answer 6

• sample space

Question 7

what does area in middle represent

Answer 7

P(A AND B)
- intersection of A and B

Question 8

what does the whole area in both circles represent

Answer 8

P(A OR B)
- union of A and B

Question 9

what does area outside of A represent

Answer 9

P(A’) = 1 - P(A)

Question 10

what does it mean if events are mutually exclusive

Answer 10

• when 2 events cannot happen at same time (events have no outcomes in common)

Question 11

using P(A) and P(B), write the statements for AND + OR that can be made for mutually exclusive events

Answer 11

• P(A AND B) = 0
• P(A OR B) = P(A) + P(B)

Question 12

if you drew a venn diagram of mutually exclusive events would the circle overlap or not

Answer 12

• they wouldn’t overlap as they cant happen at the same time

Question 13

what does it mean if events are independent. give example

Answer 13

• when one event happening doesn’t affect probability of another happening
• tossing a coin and rolling a dice are independent

Question 14

using P(A) and P(B), write the statement for AND that can be made for independent events

Answer 14

• P(A AND B) = P(A) x P(B)

Question 15

what are tree diagrams used for

Answer 15

• used to show possible outcomes for 2 or more events happening in succession

Question 16

what is a probability distribution

Answer 16

• fully describes the probability of any outcome in the sample space

Question 17

what is a random variable

Answer 17

variable whose value depends on the outcome of a random event (outcome isn’t known until experiment carried out)

(e.g when we roll 5 dice, number of 6’s rolled = r.v)

Question 18

What are the range of values the random variable can take known as. And how can it be drawn?

Answer 18

• sample space which can be drawn as a table

Question 19

the sum of the probabilities of all outcomes of an event add to what

Answer 19

one

Question 20

what is a discrete variable

Answer 20

• only takes certain countable numerical values

Question 21

what is a discrete random variable. give example

Answer 21

-values which can only take certain numerical values each of which can be assigned a probability

example: number times 1 is rolled when rolling dice 10 times (can only take certain numerical values and each can be assigned a probability)

Question 22

how can the probability distribution for a discrete random variable be described

Answer 22

• probability mass function
• table
• diagram

Question 23

in a table with little x on top row and P(X = x) on bottom row, what do these 2 things represent

Answer 23

• little x on top row: represents the event being considered and includes all possible outcomes
• P(X = x) on bottom row: represents “the probability of each event happening”

Question 24

what is a probability mass function

Answer 24

• summarises the probabilities and events which take those values

example: P(X = x) = {1/8 x=0,3 3/8 x=1,2 0 otherwise}

Question 25

give a benefit of using probability mass function

Answer 25

• impossible to list probability for every outcome, more compact

Question 26

give a benefit of using table form

Answer 26

• probability for each outcome more explicit

Question 27

for a random variable X, how can you write a statement to define probabilities add to 1

Answer 27

ΣP(X = x) = 1 for all x

Question 28

if all probabilities are the same it is known as what?

Answer 28

• a discrete uniform distribution

Question 29

what is a binomial distribution

Answer 29

• discrete probability distribution

Question 30

The discrete random variable follows a binomial distribution if it counts the number of successes when an experiment satisfies what conditions

Answer 30

• fixed number of trials (n)
• the trials are independent of each other
• two possible outcomes (success or failure)
• probability of success (p) is constant

Question 31

how can you model a random variable X as a binomial distribution

Answer 31

If X follows a binomial distribution then it is denoted X ~ B (n,p)

(n = num of trials and p = probability of success)

Question 32

if a random variable X has the binomial distribution B(n,p) then its probability mass function is given by what?

Answer 32

P(X = r) = (nCr)p∧r(1-p)∧n-r

(1-p also seen as letter q)

Question 33

what does a cumulative probability function tell you

Answer 33

• sum of all the individual probabilities up to and including chosen value