Transcript Section 6.3 ~ Probabilities With Large Numbers

Section 6.3 ~
Probabilities With Large Numbers
Introduction to Probability and Statistics
Ms. Young
Sec. 6.3
Objective
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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 gambler’s fallacy.
Sec. 6.3
The Law of Large Numbers
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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
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Ex. ~ Suppose you toss a fair coin and are interested in the probability of it
landing on heads.
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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
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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 doesn’t affect the outcome of the other trials
Sec. 6.3
Example 1
A roulette wheel has 38 numbers: 18 black numbers, 18 red
numbers, and the numbers 0 and 00 in green. (Assume that all
outcomes––the 38 numbers––have equal probability.)
a. What is the probability of getting a red number on any spin?
P(red) 
number of ways red can occur 18
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 0.474
total number of outcomes
38
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.
Sec. 6.3
Expected Value
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The expected value is the average value an
experiment is expected to produce if it is repeated a
large number of times
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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:
Expected Value = (value of event 1)(P(event 1)) +(value of event 2)(P(event 2))...
Sec. 6.3
Example 2
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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?
 1 
Expected Value = ($250)(1) + (-$100,000) 
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 500 
Expected Value = $250 - $200 = $50
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The expected profit is $50 per policy, so the expected profit for 1
million policies would be $50 million.
Sec. 6.3
Example 3
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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?
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Since there are so many events in this case, it may be easier to
create a table to find the expected value:
Sec. 6.3
Example 3 Cont’d…
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.
Sec. 6.3
The Gambler’s Fallacy
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The Gambler’s Fallacy is the mistaken belief that a
streak of bad luck makes a person “due” for a streak
of good luck
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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.
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This is known as the gambler’s fallacy, because there is not more
of a chance that you will get a heads than a tails on the next flip
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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.
Sec. 6.3
Streaks
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Another common misunderstanding that contributes to the
gambler’s 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.
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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
Sec. 6.3
Example 4
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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 hasn’t
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.