Introduction to probability

Question 1

Probability experiment - 1

Answer 1

• Procedure which produces an outcome which involves randomness

Question 2

Sample space - 2

Answer 2

• Set of all possible outcomes

- Notation: Omega Ω

Question 3

Event - 3

Answer 3

• Collection of outcomes
• Notation: Capital letters
• • Example: A = {even number is thrown} = {2,4,6}

Question 4

Simple event - 1

Answer 4

• Event consisting of only one outcome

Question 5

Probability measure - 2

Answer 5

• Function P(x)

- Assigns value between 0-1 to an event x

Question 6

Interpretation of probabilities - 3

Answer 6

• P(x) = 0 [Event is impossible]
• P(x) = 1 [Event is certain]
• P(x) < 0.05 [Event is unlikely]

Question 7

How to determine P(x) - 3

Answer 7

• Relative frequency
• Classical theoretical approach
• Subjective approach

Question 8

Relative frequency - 1

Answer 8

• P(x) = Num of times x occurred / number of times procedure is repeated

Question 9

Classical theoretical approach - 3

Answer 9

• Theoretical outcome calculated by making probability model
• Dice rolls, card games…

Question 10

Subjective approach - 1

Answer 10

• Calculation of P(x) based on intuition and experience

Question 11

LLN, Law of Large Numbers - 2

Answer 11

• Suppose: If procedure is repeated again and again and outcomes do not depend on previous outcomes
• Then: Relative frequency of probability of event A tends to approach actual probability P(x)

Question 12

Classical approach - 1

- - Determining P(x) if all outcomes are equally likely

Answer 12

• If all outcomes are equally likely then the probability is:
  Num of ways x can occur / Total num of different simple events

Question 13

Counting principle - 2

Answer 13

• If first experiment has a ≥ 0 possible outcomes and for each outcome of the first experiment the second has b ≥ 0 outcomes, the two experiments combined have a x b possible outcomes.
• This principle extends to any number of experiments.

Question 14

General probability measure for finite sample space - 3

Answer 14

• Ω is finite (or countable)
• Each outcome ω belonging to Ω has a probability P(ω) assigned, which ω ≥ 0, and the sum of all P(ω) = 1
• The probability of event P(A) is equal to the sum of all probabilities in which ω belongs to event A.

Question 15

How to find P(A)

Answer 15

• Define sample space Ω
• Determine probabilities for each ω in Ω, if all outcomes are equally likely then P(ω) = 1/N
• Determine which outcomes belong to A
• Compute P(A) by Summing all Probabilities of outcomes which belong to A