Law of Numbers

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The law of large numbers (LLN) explains that as the number of trials increase, the random variable becomes stable in the long-run. If a trial of probable situation is repeated again and again, then the more times the trial is repeated, the more likely it is that the frequency of any particular event will be close to the probability of that event.


If a coin is tossed many times, the more times it is tossed, the likelihood of the number of "heads" in the total population will be close to 1/2. This Law of Large Numbers can be further explained with the help of a Randomly Generated Coin Toss online applet (available from
The coin is unbiased and it has two sides that are equally likely to come up. When the random generator is run, the applet shows the proportion of heads in the total population. In the first 10 tosses the proportion of heads is 0.272 (3 heads and 7 tails). When it is run for a longer time up to 100 tosses the proportion of heads approaches one-half and becomes 0.48 (43 heads and 47 tails). For a 1000 tosses the proportion of heads become 0.499 (502 heads and 498 tails). This figure will fluctuate around 0.5, with the fluctuations slowly getting smaller and almost reaching 0.5. [1]
1) Let's say you flipped the coin once and it landed on heads. You will expect that on alternative tosses you will get a head. In 10 coins are tossed you expect 5 to be heads since the expected percentage of successes is 50%. But in reality only three are heads. The difference between the actual and expected number of successes is 2. The actual percentage of number of heads is 20% meaning a difference between actual and expected percentage of 30%. ...
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