Section 6.3 ~ Probabilities With Large Numbers

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Section 6.3 Probabilities With Large Numbers

• Introduction to Probability and Statistics
• Ms. Young

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Objective
Sec. 6.3

• After this section you will understand the law of
  large numbers, use this law to calculate expected
  values, and recognize how misunderstanding of the
  law of large numbers leads to gamblers fallacy.

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The Law of Large Numbers
Sec. 6.3

• Recall that the C.L.T. states that as the sample
  size increases, the sample mean will approach the
  population mean and the sample standard deviation
  will approach the population standard deviation
• Simply put, the law of large numbers (or law of
  averages) states that conducting a large number
  of trials will result in a proportion that is
  close to the theoretical probability
• Ex. Suppose you toss a fair coin and are
  interested in the probability of it landing on
  heads.
• The theoretical probability is 1/2, or .5, but
  tossing the coin 10 times may result in only 3
  heads resulting in a probability of .3.
• Tossing a coin a 100 times on the other hand,
  will result in a probability much closer to the
  theoretical probability of .5.
• And tossing a coin 10,000 times will be even
  closer to the theoretical probability
• You can think of the law of large numbers like
  the central limit theorem, the larger the sample
  size, the closer you get to the true probability
• Keep in mind though, that the law of large
  numbers only applies when the outcome of one
  trial doesnt affect the outcome of the other
  trials

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Example 1
Sec. 6.3

• A roulette wheel has 38 numbers 18 black
  numbers, 18 red
• numbers, and the numbers 0 and 00 in green.
  (Assume that all
• outcomesthe 38 numbershave equal
  probability.)
• a. What is the probability of getting a red
  number on any spin?
• b. If patrons in a casino spin the wheel
  100,000 times, how many times
• should you expect a red number?
• The law of large numbers tells us that
  as the game is played more
• and more times, the
  proportion of times that a red number appears
• should get closer to
  0.474. In 100,000 tries, the wheel should
• come up red close to
  47.4 of the time, or 47,400 times.

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Expected Value
Sec. 6.3

• The expected value is the average value an
  experiment is expected to produce if it is
  repeated a large number of times
• Because it is an average, we should expect to
  find the expected value only when there are a
  large number of events, so that the law of large
  numbers comes into play
• The following formula is used to calculate
  expected value

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Example 2
Sec. 6.3

• Suppose the InsureAll Company sells a special
  type of insurance in which it promises you
  100,000 in the event that you must quit your job
  because of serious illness. Based on past data,
  the probability of the insurance company having
  to payout is 1/500. What is the expected profit
  if the insurance company sells 1 million policies
  for 250 each?
• The expected profit is 50 per policy, so the
  expected profit for 1 million policies would be
  50 million.

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Example 3
Sec. 6.3

• Suppose that 1 lottery tickets have the
  following probabilities 1 in 5 win a free
  ticket (worth 1), 1 in 100 win 5, 1 in 100,000
  win 1,000, and 1 in 10 million win 1 million.
  What is the expected value of a lottery ticket?
• Since there are so many events in this case, it
  may be easier to create a table to find the
  expected value

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Example 3 Contd
Sec. 6.3
The expected value is the sum of all the products
(value probability), which the final column of
the table shows to be 0.64. Thus, averaged
over many tickets, you should expect to lose 64
for each lottery ticket that you buy. If you buy,
say, 1,000 tickets, you should expect to lose
about 1,000 0.64 640.
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The Gamblers Fallacy
Sec. 6.3

• The Gamblers Fallacy is the mistaken belief that
  a streak of bad luck makes a person due for a
  streak of good luck
• Ex. The odds of flipping a coin so that it
  comes up heads 20 times in a row, assuming the
  coin is fair, are extremely low, 1/1,048,576 to
  be exact. Therefore, if you have flipped a coin
  and it has come up 19 times in a row, many people
  would be eager to lay very high odds against the
  next flip coming up tails.
• This is known as the gamblers fallacy, because
  there is not more of a chance that you will get a
  heads than a tails on the next flip
• Once the 19 heads have already been flipped, the
  odds of the next flip coming up tails is still
  just 1 in 2. The coin has no memory of what has
  gone before, so although it would be extremely
  rare to come up with 20 heads in a row, the 20th
  toss still just has a 50/50 chance of landing on
  heads or tails.

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Streaks
Sec. 6.3

• Another common misunderstanding that contributes
  to the gamblers fallacy involves expectations
  about streaks
• Ex. Suppose you toss a coin 6 times and see the
  outcome to be HHHHHH and then you toss it six
  more times and see the outcome to be HTTHTH.
• Most people would look at these outcomes and say
  that the second one is more natural and that the
  streak of heads is surprising
• Since the possible number of outcomes is 64 (26
  64), each individual outcome has the same
  probability of 1/64

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Example 4
Sec. 6.3

• A farmer knows that at this time of year in his
  part of the country, the probability of rain on a
  given day is 0.5. It hasnt rained in 10 days,
  and he needs to decide whether to start
  irrigating. Is he justified in postponing
  irrigation because he is due for a rainy day?
• The 10-day dry spell is unexpected, and, like a
  gambler, the farmer is having a losing streak.
  However, if we assume that weather events are
  independent from one day to the next, then it is
  a fallacy to expect that the probability of rain
  is any more or less than 0.5.