Section 7C: The Law of Large Numbers

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1 Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half the time and tails the other half of the time. So if a coin is being flipped 00 times, one could expect about 00 = 0 heads and about 00 = 0 tails. Suppose the coin is unfair, and the probability of getting heads is 3. How many times would you expect 4 to get heads? tails? One would expect this coin to come up heads three-quarters of the time and tails the other quarter of the time. So if this coin is being flipped 00 times, one could expect about = 7 heads and about 4 00 = tails. The Law of Large Numbers (Law of Averages) The law of large numbers applies to a process for which the probability of an event A is P (A) and the results of repeated trials are independent. It states: If the process is repeated over 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 the P (A). Example. You roll a fair six-sided standard die 300 times. How many times would you expect the value to be? Since the probability of getting a on a die roll is 6, one would expect that a would come up about one sixth of the time, or = 0 times. How many times would you expect the value to be odd? The probability of rolling an odd value is 3 6 =, so one would expect to roll an odd value half of the time, or 300 = 0 times. If you want to have the value 4 come up roughly 00 times, how many times should you expect you must roll the die? The probability of rolling a 4 is 6, so to achieve 00 rolls which have a value of 4, we must answer the following question: How many rolls must be made so that one sixth of them is roughly 00? ( 6? = 00) Solving this gives that we would need about 00 rolls to get roughly 00 rolls with a value of 4.

2 Example. You play a game in which you flip a coin. If the coin comes up heads, you receive point. If the coin comes up tails, you receive points. You flip the coin 00 times. How many times would you expect to get heads? tails? As in the last example, one would expect about 00 = 0 heads and 00 = 0 tails. How many points would you expect to win in these 00 flips? Using the information from the previous part, if we expect 0 heads, we will receive 0 = 0 points from the heads. If we expect 0 tails, we will receive 0 = 00 points, making for an expected total of 0 points. Expected Value Consider two events, each with its own value and probability. The expected value is ( ) ( ) ( ) ( event event event event expected value = + value probability value probability ) This formula can be extended to any number of events by including more terms in the sum. Example. Refer to the example above. Find the expected point value of a single coin flip. We can analyze this with a table with three columns: one listing the possible events, one giving the probability of each event, and one giving the value associated to each event. Getting Heads point Getting Tails points Next, we will multiply the probability of each event by their associated value. We then add up those quantities; the result is the expected value. This can be done nicely along-side the table: Getting Heads point point = point Getting Tails points points = point + = 3 =. points So the expected value of a single coin flip is. points. Use this to compute the expected point total for 00 flips. Multiply the expected value by the number of trials to get the expected value afterward:. 00 = 0 points. Notice this is the same as we computed earlier in the problem. How many points can you expect after 00 flips? 000 flips? We can repeat the previous method to get an expected value of. 00 = 300 points after 00 flips and. 000 = 00 points after 000 flips.

3 Example. A lottery for a particular state costs $ and has the payouts and probabilities listed below: Prize Probability $,000,000 8,00,000 $0,000 00,000 $00 0,000 $ Find the expected value for the payout of a single lottery ticket. We will multiply each event s probability by its value, then add those quantities: Prize Probability $,000,000 8,00,000 $, 000, 000 8,00,000 = 37 $0.0 $0,000 00,000 $0, ,000 = $0.0 $00 0,000 $00 0,000 = $0.0 $ $ = $0.0 $, 000, 000 8,00,000 + $0, ,000 + $00 0,000 + $ = $0.8 So the expected value of the return of a lottery ticket is $0.8. (Note, however, that you must spend $ in order to get any return.) What is the expected winnings if you purchase one ticket every day for a year? If you play 36 times, your expected winnings would be $ $0.0. What is your net gain in that year? In that year, a total of $36 was spent while earning only $0.0, so the net gain is $0.0 (amount gained) $36 (amount spent) = $6.80 There is a net loss of $6.80 by playing the lottery in this manner for a full year. 3

4 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. Example. You play a game by flipping a coin. You receive $ for each head and you lose $ for each tail. After 00 flips, say you have 40 heads and 60 tails. What is your net gain thus far in the game? You won $40 from the heads and lost $60 from the tails, so there is a net gain of $60, or a net loss of $60. What is the empirical probability of getting heads based on your first 00 flips? Based on these 00 flips, the empirical probability of getting heads is = = 0.4. Now, suppose you play another 00 times (for a total of 00 times) and get 4 heads and tails. What is your net gain after all 00 flips? Now, we have a total of 8 heads and tails. So we won $8 and lost $, making for a net gain of $8 $ = $30, or a net loss of $30. What is the empirical probability of getting heads based on these 00 flips? Based on these 00 flips, the empirical probability of getting heads is 8 00 = 0.4. Does this result agree with the Law of Large Numbers? This seems counter-intuitive, because even though the probability of getting heads is getting closer to half (as one would expect), the loss in playing the game is getting larger. This is the mistake many make when quoting the law of large numbers. All this law tells you is that after time, the empirical probability should get close to the theoretical probability. It does not, however, ensure that losses will be recouped in any way. In fact, it is possible for the losses to get arbitrarily large if you play the game long enough. 4

5 Example. An American roulette table has a wheel with slots. The wheel is spun and a ball is released. Players bet on in which of the slots the ball with finally land. There are 8 black slots, 8 red slots, and green slots. What is the probability of getting red on a roulette spin? There are 8 red spots on the roulette wheel out of a total of spots. probability of landing on red is 8 = Assuming they are all equally likely, the If you make a bet on red and the ball lands in a red slot, you get back your bet doubled. What is the expected value of your return if you bet $0 on red? There are two outcomes worth noting when you are betting on red, and those are landing on red and not landing on red. As indicated in the previous question, there is a 8 probability that you will land on red. Therefore, there is a 8 = 0 probability of not landing on red. If you land on red, you will make $0, and if you do not, you earn nothing back. We can make a table for these events with their probabilities and values, then compute the expected value from there: 8 8 Landing on Red $0 $0 $ Not Landing on Red $0 $0 = $0 8 0 $0 + $0 $9.47 Therefore, the expected winnings on one spin is $9.47. (Note that you must pay $0 to spin, so there is a net loss of $0.3 on each spin!) What would be your expected winnings if you played 0 times? Multiply the expected winnings for one spin ($9.47) by 0 to get your total winnings: $ = $ What is your net gain (or loss) after playing 0 times? 00 times? It costs 0 $0 = $00 to play roulette 0 times, so the net gain is $00 = $6.0. Note that we could have found this by taking the expected value with the cost of the spin ($ , make sure to retain as many digits as possible) and multiply it by 0. Similarly, we can take the net expected value and multiply it by 00 to get a net loss of $ = $.63 after playing 00 times. What is the moral of the story? The house always wins in the long run.