Statistical Sciences 2244A/B Chapter Notes - Chapter 9: Sample Space, Formal Language, Randomness
22 May 2018
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Stats 2244
Chapter 9
CHAPTER 9.1
The Idea of Probability
- Chance behavior is unpredictable in the short run but has a regular and predictable pattern in
the long run
- Note: don’t confuse random and random sampling!!
o Random sampling = selecting individuals from a population such that every individual in
the population has the same chance of being selected
o Randomness = outcomes are determined by chance (but the probabilities do not have to
be the same for all outcomes)
- We call something random if individual outcomes are uncertain but there is a regular
distribution of outcomes in a large # of repetitions
- The probability of any outcome of a random phenomeon is the proportion of times the outcome
would occur in a very long series of repetitions
- Probability describes only what happens in the long run – this is called a frequentist approach to
defining probabilities (bc we rely on the relative frequency (proportion) of one particular
outcome among very many observations of the random phenomenon)
CHAPTER 9.2
Probability Models
- The sample space, S of a random phenomenon is the set of all possible outcomes
- An event is an outcome or a set of outcomes of a random phenomenon
o An even is a subset of a sample space
- A probability model is a mathematical description of a random phenomenon consisting of 2
parts: a sample space, S and a way of assigning probabilities to events
CHAPTER 9.3
Probability Rules
1. Any probability is a # bw 0 and 1, inclusively
o An event with probability 0 never occurs
o An event with probability 1 occurs on every trial
o An event with probability 0.5 occurs in half the trials in the long run
2. All possible outcomes together must have probability 1
3. If 2 events have no outcomes in common, the probability that one or the other occurs is the sum
of their individual probabilities
o If one event occurs in 40% of all trials, a different event occurs in 25% of all trials, and
the two can never occur together, then one or the other occurs on 65% of all trials
because 40% + 25% = 65%.
4. The probability that an event does not occur is 1 minus the probability that the event does occur
o If an event occurs in (say) 70% of all trials, it fails to occur in the other 30%. The
probability that an event occurs and the probability that it does not occur always add to
100%, or 1.
- Capital letters near the beginning of the alphabet denote events. If A is any event, we write its
probability as P(A).
- Those 4 runs in formal language:
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. We call something random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large # of repetitions would occur in a very long series of repetitions. The probability of any outcome of a random phenomeon is the proportion of times the outcome. The sample space, s of a random phenomenon is the set of all possible outcomes. An event is an outcome or a set of outcomes of a random phenomenon: an even is a subset of a sample space. A probability model is a mathematical description of a random phenomenon consisting of 2 parts: a sample space, s and a way of assigning probabilities to events. If 2 events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.
