STAT2003 Chapter Notes - Chapter 12-13: Bayes Estimator, Slot Machine, Empirical Probability
Ch 12. From Randomness to Probability, Ch 13. Probability Rules!
Covers: 10/9, 10/11, 10/13, 10/16, 10/18, 10/20 (6 lectures, Weeks 7 & 8)
Ch 12. From Randomness to Probability
Random Phenomena
● Ex. Red Light, Green Light
○ Every day one drives through an intersection and checks if the light is red, green
or yellow
■ Day 1: green
■ Day 2: red
■ Day 3: green
○ Before beginning, the following are known:
■ The possible outcomes and that an outcome will occur
○ After finishing (at the stoplight), it is known:
■ The outcomes that occurred
● Some Vocab:
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○ Trial: Each occasion which we observe a random phenomena
○ Outcome: The value of the trial for the random phenomena
○ Event: The oiatio of the tial’s outoes
○ Sample Space: The collection of all possible outcomes
● An Ex. with the vocab: Flipping Two Coins
○ Tial → The flippig of the to ois
○ Outoe → Heads o tails fo eah flip
○ Eet → HT, fo eaple
○ “aple “pae: → S = {HH, HT, TH, TT}
● Again, The Law of Large Numbers
○ If you flip a coin once, you will either get 100% heads of 0% heads.
○ If you flip a coin 1000 times, you will probably get much closer to 50% heads
○ This LLN states that for many trials, the proportion of times an event occurs
settles down to one number.
○ This number is called the empirical probability
○ Requirements:
■ Identical Probabilities:
● The probabilities for each event must remain the same for each
trial.
■ Independence:
● The outcome of a trial is not influenced by the outcomes of the
previous trials
■ Empirical probability
● #
# (in the long run)
○ Using the Law of Large Numbers for the Red Light, Green Light Scenario
■ After several days, the proportion of green lights encountered is
approximately 0.35 (as indicated by the plateauing/leveling off of the
graph above)
■ P(green) = 0.35
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■ If more days were recorded, the probability would still be about 0.35
○ The Nonexistent Law of Averages?
■ If you flip a coin 5 times and get five tails, then you are due for a head on
the next flip.
■ You put 10 quarters in the slot machine and lose each time. You are just
a bad luck person, so you have a smaller chance of winning on the 11th
try.
■ Ex. suppose a couple has 3 children all of whom are boys, is the couple
more likely to have a girl for the next child?
■ There is no such thing as the Law of Averages for short runs.
● Theoretical Probability
○ Ex. American Roulette (game)
■ 18 Red, 18 Black, 2 Green
■ If you bet on Red, what is the probability of winning?
■ Theoretical Probability
■ #
#→
○ Ex. Heads or Tails
■ Flip 2 coins. Find P(HH)
● List the sample space:
● S = {HH, HT, TH, TT}
● P(HH) = ¼
■ Flip 100 coins. Find the probability of all heads.
● The sample space would involve
1,267,650,600,228,229,401,496,703,205,376 different outcomes.
● There is a better way to do this.
● Equally Likely?
○ What’s og ith this logi? Let’s see →
■ Randomly pick two people.
■ Find the probability that both are left-handed.
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Document Summary
Covers: 10/9, 10/11, 10/13, 10/16, 10/18, 10/20 (6 lectures, weeks 7 & 8) Every day one drives through an intersection and checks if the light is red, green or yellow. The possible outcomes and that an outcome will occur. After finishing (at the stoplight), it is known: Trial: each occasion which we observe a random phenomena. Outcome: the value of the trial for the random phenomena. Event: the (cid:272)o(cid:373)(cid:271)i(cid:374)atio(cid:374) of the t(cid:396)ial"s out(cid:272)o(cid:373)es. Sample space: the collection of all possible outcomes. An ex. with the vocab: flipping two coins. T(cid:396)ial the flippi(cid:374)g of the t(cid:449)o (cid:272)oi(cid:374)s. Out(cid:272)o(cid:373)e heads o(cid:396) tails fo(cid:396) ea(cid:272)h flip. A(cid:373)ple pa(cid:272)e: s = {hh, ht, th, tt} If you flip a coin once, you will either get 100% heads of 0% heads. If you flip a coin 1000 times, you will probably get much closer to 50% heads.
