ECON 1011 Lecture Notes - Lecture 4: Sample Space, Randomness, Fair Coin
16 May 2018
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Randomness and Probability
Chapter 5
Random Phenomena and Probability
With random phenomena, we cannot predict the individual outcomes, but we can hope to understand
characteristics of their long-run behaviour.
For any random phenomenon, each attempt, or trial, generates an outcome.
We use the more general term event to refer to outcomes or combination of outcomes.
Random Phenomena and Probability
The sample space is the collection of all possible outcomes.
We deote the saple spae “ o soeties Ω.
Take a coin toss as an example: For a single coin toss: S = {H, T} For two coin tosses: S = {HH, HT, TH, TT}
Random Phenomena and Probability
The probability of an event is its long-run relative frequency.
Going back to our coin toss example, while we may not be able to predict whether we get heads or tails, we
can say that in the long run the percentage of heads is about 50%.
This is the probability of getting a head in a coin toss.
Random Phenomena and Probability
The concept of probability is drawn from the idea of symmetrical outcomes.
The book refers to this as a Model-Based Theoretical Probability.
Fo istae, hat is the poailit of gettig a he thoig a si-sided die?
Suppose you throw 2 dice. What is the probability that the sum of the two dice will be 6?
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In some instances, probability is best thought of as subjective.
This is sometimes called personal probability.
You a ask a peso this uestio: What is the poailit that Donald Trump will win the Republican party
pesidetial piaies?
The ase to this uestio eflets a idiidual’s pesoal opiio.
Independence means that the outcome of one trial does not influence or change the outcome of another.
Two events, A and B, are independent events if the probability of Event B occurring is the same whether or
not Event A occurs.
For example, if a fair coin is tossed two times, the probability that we get a head on the second toss is ½,
regardless of whether or not we get a head in the first toss.
Consider the two events: (1) It will rain tomorrow in Malvern. (2) It will rain tomorrow in Wattle Park. Are
these independent?
The Law of Large Numbers (LLN) states that if the events are independent, then as the number of trials
increases, the long-runrelative frequency of an event gets closer and closer to a single value.
Probability Rules
Rule #1
If the poailit of a eet ouig is , the eet a’t ou.
If the probability is 1, the event always occurs.
For any event A,
Rule #2: The Probability Assignment Rule
The probability of the set of all possible outcomes must be 1.
where S represents the set of all possible outcomes and is called the sample space.
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Rule #3: The Complement Rule
The probability of an event occurring is 1 minus the probability that it does’t ou.
hee the set of outoes that ae ot i eet A is alled the opleet of A, ad is deoted Ac
Rule #4: The Multiplication Rule
For two independent events A and B, the probability that both A and B occur is the product of the
probabilities of the two events.
provided that A and B are independent.
For example, if we flip a coin twice, what is the probability it will come up heads both times?
Event A is that the coin comes up heads on the first flip, and Event B is that the coin comes up heads on the
second flip.
Since P(A) and P(B) equal 1/2, the probability that both events occur, P(A and B), is P(A) x P(B) = ¼.
What if you flip a coin and roll a six sided die? What is the probability that the coin comes up head, and the
die comes up 1?
Rule #5: The Addition Rule
Two events are disjointed (or mutually exclusive) if they have no outcomes in common.
The Addition Rule allows us to add the probabilities of disjoint events to get the probability that either event
occurs.
where A and B are disjoint.
For example, if a 6-sided die is rolled. What is the probability of rolling a 2 or a 5?
Event A: The number rolled is a 2.
Event B: The number rolled is a 5.
The events are mutually exclusive since they cannot occur at the same time. Hence P(A or B) = P(A) + P(B) =
1/6 + 1/6 = 1/3.
Suppose a glass jar contains 1 red, 3 green, 2 blue, and 4 yellow marbles. If we choose a single marble from
the jar, what is the probability that it is yellow or green?
Rule #5: The General Addition Rule
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Document Summary
With random phenomena, we cannot predict the individual outcomes, but we can hope to understand characteristics of their long-run behaviour. For any random phenomenon, each attempt, or trial, generates an outcome. We use the more general term event to refer to outcomes or combination of outcomes. The sample space is the collection of all possible outcomes. We de(cid:374)ote the sa(cid:373)ple spa(cid:272)e o(cid:396) so(cid:373)eti(cid:373)es . Take a coin toss as an example: for a single coin toss: s = {h, t} for two coin tosses: s = {hh, ht, th, tt} The probability of an event is its long-run relative frequency. Going back to our coin toss example, while we may not be able to predict whether we get heads or tails, we can say that in the long run the percentage of heads is about 50%. This is the probability of getting a head in a coin toss. The concept of probability is drawn from the idea of symmetrical outcomes.
