Basics of Probability & Models (Exam 2)

Question 1

phenomenon

Answer 1

• some result we would like to observe
• chance behavior is unpredictable in the short run, but it has predictable pattern in the long run
• random doesn’t mean that everything is equally likely, just means that it’s predictable

Question 2

sample space (S)

Answer 2

• the set of ALL possible outcomes of a phenomenon, such that the outcomes are mutually exclusive
• ex: flipping a coin → S = {H, T}
• ex: tossing a dice → S = {1, 2, 3, 4, 5, 6}
• can either be discrete or continuous

Question 3

discrete (sample space)

Answer 3

• when the total # of possible outcomes can be easily defined and written out
• positive, full numbers
• use {squiggly brackets}
• ex: coin/dice

Question 4

continuous (sample space)

Answer 4

• when the total number of possible outcomes is uncountable and infinite and can’t be written (we have an idea of upper & lower bounds, but so many decimals inbetween)
• ex: student’s height down to the fraction of an inch
• use [straight brackets]

Question 5

event

Answer 5

• any collection of outcomes of a random phenomenon
• ex: the result of a dice toss:
  Event A: the result is no greater than 3
  A = {1, 2, 3}
  Event B: the result is even
  B = {2, 4, 6}
  Event C: the result is 4
  C = {4}

Question 6

probability

Answer 6

• attempts to describe the long-term patterns of a random phenomenon
• probability of an event is the proportion of times the event occurs in many repeated trials of a random phenomenon
• ex: flip a coin 1,000 and see 500 heads
  The probability of heads, P[coin = heads] = 0.5
  The probability of tails, P[coin = tails] = 0.5

Question 7

independence

Answer 7

• the outcome of one trial does NOT influence the outcome of any other trial
• ex: if you flip heads the first time, the next coin flip is still 50/50 if you get heads or tails
• we want true independence in repeated trials, or a very large population size so sampling has a minimal effect

Question 8

probability models

Answer 8

• a way of structuring our knowledge about random phenomenon
• can be discrete or continuous

Question 9

complement

Answer 9

• the event that it does NOT occur
• complement of some event A is written as A^c (event that A does not occur)
• a partition of our sample space (a perfect division)… it’s either an event A or its not (a complement A^c)… either satisfies or doesn’t satisfy event
• A + A^c = S

Question 10

disjoint

Answer 10

• if events don’t share any outcomes
• can NOT occur simultaneously

Question 11

probability axioms

Answer 11

defining our probability models
1) 0 < P(A) < 1
- Probabilities of events are always between 0 and 1
2) P(S) = 1
- Probability that the outcome in the sample space is = 1
3) P(A^c) = 1 – P(A)
- The probability that A does not occur is
4) P(A or B) = P(A) + P(B)
- If A & B are disjoint

Question 12

discrete probability models

Answer 12

• a probability model assigns a probability to every possible outcome (every probability must be 0-1)
• the sum of all outcome probabilities must = 1
• to calculate probability of an event, sum the probabilities of the outcomes

Question 13

continuous probability models

Answer 13

• probability of any individual outcome = 0
• there are so many possible outcomes that any one outcome is extremely unlikely to occur
• take the probability of ALL intervals
• probabilities are assigned density curves

Question 14

density curves

Answer 14

• for continuous probability models
• RULE 1: no part of a density curve can be negative
• RULE 2: the total area under the curve must = 1