Examining the Law of the Numbers

The law of large numbers says that in repeated, independent trials with the same probability p of success in each trial, the chance that the percentage of successes differs from the probability p by more than a fixed positive amount, e> 0, converges to zero as the number of trials n goes to infinity, for every positive e" (Stark, 2006).

The law of large numbers means that if you do X enough times you will succeed. The law of large numbers states that the more an experiment or test is done, the more likely the outcome will approach it's theoretical probability.

AS the sample size gets larger, the percentage of each event will get closer and closer to the expectancy. If we were flipping a coin, the expectancy is 50% 'heads' and 50% 'tails'. In a sample of 10 tosses, I expect 5 heads and 5 tails, had 7 heads and 3 tails. However, as my example size grew, I got closer and closer to 50%, But the number which one side leads the other will increase. For example, at 100,000 tosses I was at 49.5% heads and 50.5% tails, thus tails lead heads by 1% or an actual number of 1000. At 1,000,000 tosses, I was at 49.75% heads and 50.25% tails, but tails has actually come up 50,000 more times than heads. At what point does it even up? That's the question you're asking and the answer is: It may never hit exactly 50-50.

Using the data provided, with 49 heads from 100 tosses, and 101 heads from 200 tosses. The difference in the number of heads decreases from 1 to 2. The percentage difference also decreases . See chart head or tails.

Another example of the Law of Large Numbers is tossing a coin 5 times,. Lets us H for heads and T for tails. The out come could be all H or all T. There is a 50% or 1/5 chance that it will land of H or T. If we toss the coin 10 times are chance of landing on H increase because we have more chance to do so.

We use a coin toss, because the initial probability of tossing a head, or tail, when flipping the coin is 50 percent. According to the law of large numbers, if you flip a coin a thousand, or even a million times, it's heads, or tails flipped will be close to 50 %. If a coin is tossed 1,000 times you have a 50% of getting 500 heads. If the coin is fair it is most likely that you will get 500 heads and 500 tails. But because of variation this may not happen.

If you toss a coin 5 times and get tails the first 3 times you still have a 50% of getting either heads or tails the next two tosses. The coin has only two sides, each side has an equal chance. According to the law of large numbers the more we toss the coin the more likely it is to even out. So to test this I tossed a penny 5 times and got heads 4 times and tails 1. My goal is to to have an equal numbers of heads and an equal number of tails. After 10 tosses there were still more heads then tails. See heads or tails chart. After tossing the coin 30 times I finally reached an equal number of heads and equal number of tails. By looking at at the heads or tails chart we can see that throughout the experiment heads would over power the number of tails and the tails would do the same.

References

Stark., (2006). Law of large numbers. Retrieved December 7, 2006, from Statistics Tools for

Internet and Classroom Instruction Web site:

http://www.stat.berkeley.edu/~stark/Java/Html/lln.htm