1. Section 6.3 ~ Probabilities With Large Numbers
Introduction to Probability and Statistics
Ms. Young

2. Sec. 6.3
Objective
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.

3. The Law of Large Numbers
Sec. 6.3
The Law of Large Numbers
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 doesn’t affect the outcome of the other trials

4. 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?
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 In 100,000 tries, the wheel should
come up red close to 47.4% of the time, or 47,400 times.

5. Sec. 6.3
Expected Value
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:

6. Sec. 6.3
Example 2
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.

7. Sec. 6.3
Example 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:

8. 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.

9. Sec. 6.3
The Gambler’s Fallacy
The Gambler’s 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 gambler’s 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.

10. Sec. 6.3
Streaks
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.
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

11. Sec. 6.3
Example 4
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.