1. 6.3 Probabilities with Large Numbers
LEARNING GOAL
Understand the law of large numbers, use this law to understand and calculate expected values, and recognize how misunderstanding of the law of large numbers leads to the gambler’s fallacy.
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2. The Law of Large Numbers
The law of large numbers (or law of averages) applies to a process for which the probability of an event A is P(A) and the results of repeated trials do not depend on results of earlier trials (they are independent).
It states: If the process is repeated through many trials, the proportion of the trials in which event A occurs will be close to the probability P(A). The larger the number of trials, the closer the proportion should be to P(A).
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3. a. What is the probability of getting a red number on any spin?
EXAMPLE 1 Roulette
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?
Solution:
a. The theoretical probability of getting a red number on any spin is
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P(A) = = = 0.474
number of ways red can occur
total number of outcomes 18 38
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4. EXAMPLE 1 Roulette
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.)
b. If patrons in a casino spin the wheel 100,000 times, how many times should you expect a red number?
Solution:
b. 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 In 100,000 tries, the wheel should come up red close to 47.4% of the time, or about 47,400 times.
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5. Expected Value Definition
The expected value of a variable is the weighted average of all its possible events. 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.
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6. Calculating Expected Value
Consider two events, each with its own value and probability. The expected value is
expected value =
(value of event 1) * (probability of event 1)
+ (value of event 2) * (probability of event 2)
This formula can be extended to any number of events by including more terms in the sum.
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7. EXAMPLE 2 Lottery Expectations
Suppose that $1 lottery tickets have the following probabilities: 1 in 5 to win a free ticket (worth $1), 1 in 100 to win $5, 1 in 100,000 to win $1,000, and 1 in 10 million to win $1 million. What is the expected value of a lottery ticket? Discuss the implications. (Note: Winners do not get back the $1 they spend on the ticket.)
Solution: The easiest way to proceed is to make a table (next slide) of all the relevant events with their values and probabilities. We are calculating the expected value of a lottery ticket to you; thus, the ticket price has a negative value because it costs you money, while the values of the winnings are positive.
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8. EXAMPLE 2 Lottery Expectations
Event
Value
Probability
Value × probability
Ticket purchase
Win free ticket
Win $5
Win $1000
Win $1 million
-$1 $1 $5
$1,000
$1,000,000 1
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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9. TIME OUT TO THINK
Many states use lotteries to finance worthy causes such as parks, recreation, and education. Lotteries also tend to keep state taxes at lower levels. On the other hand, research shows that lotteries are played by people with low incomes. Do you think lotteries are good social policy? Do you think lotteries are good economic policy?
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10. The Gambler’s Fallacy Definition
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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11. EXAMPLE 3 Continued Losses
You are playing the coin-toss game in which you win $1 for heads and lose $1 for tails. After 100 tosses, you are $10 in the hole because you have 45 heads and 55 tails.
You continue playing until you’ve tossed the coin 1,000 times, at which point you’ve gotten 480 heads and 520 tails.
Is this result consistent with what we expect from the law of large numbers? Have you gained back any of your losses? Explain.
Solution: The proportion of heads in your first 100 tosses was 45%. After 1,000 tosses, the proportion of heads has increased to 480 out of 1,000, or 48%.
Because the proportion of heads moved closer to 50%, the results are consistent with what we expect from the law of large numbers.
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12. EXAMPLE 3 Continued Losses
Solution: (cont.)
However, you’ve now won $480 (for the 480 heads) and lost $520 (for the 520 tails), for a net loss of $40.
Thus, your losses increased, despite the fact that the proportion of heads grew closer to 50%.
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13. Streaks
Another common misunderstanding that contributes to the gambler’s fallacy involves expectations about streaks.
Suppose you toss a coin six times and see the outcome HHHHHH (all heads). Then you toss it six more times and see the outcome HTTHTH.
Most people would say that the latter outcome is “natural” while the streak of all heads is surprising.
But, in fact, both outcomes are equally likely. The total number of possible outcomes for six coins is 2 × 2 × 2 × 2 × 2 × 2 = 64, and every individual outcome has the same probability of 1/64.
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14. Moreover, suppose you just tossed six heads and had to bet on the outcome of the next toss. You might think that, given the run of heads, a tail is “due” on the next toss.
But the probability of a head or a tail on the next toss is still 0.50; the coin has no memory of previous tosses.
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15. TIME OUT TO THINK
Is a family with six boys more or less likely to have a boy for the next child? Is a basketball player who has hit 20 consecutive free throws more or less likely to hit her next free throw? Is the weather on one day independent of the weather on the next (as assumed in the next example)? Explain.
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16. EXAMPLE 4 Planning for Rain
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?
Solution: 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.
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17. The End
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