Suppose you toss a fair coin 10,000 times. Should you expect to get exactly 5,000 heads? Why or why not? What does the law of large numbers say about the results you are likely to get? a. You shouldn't expect to get exactly 5,000 heads, as you cannot predict the exact number of heads that will occur. b. You should expect to get approximately 5,000 heads because the proportion of heads should be around 50% for such a large number of tosses. c. You should expect to get approximately 5,000 heads because, with a fair coin, the expected proportion of heads is 50%. d. You shouldn't expect to get exactly 5,000 heads, because precisely counting the number of heads that occur is challenging. Correct Answer: The law of large numbers implies that as the number of coin tosses increases, the actual proportion of heads is very likely to get close to the expected proportion, which is 50%. However, getting exactly 5,000 heads is not guaranteed because each toss is independent, and there is an inherent variability in any random process. Hence the most accurate statement is b, which correctly reflects that you should expect to get approximately 5,000 heads, with the understanding that "approximately" accounts for the likely slight deviation from the exact number due to chance.

b. You should expect to get approximately 5,000 heads because the proportion of heads should be around 50% for such a large number of tosses.

Extra: The Law of Large Numbers is a principle of probability that describes the result of performing the same experiment a large number of times. According to this law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer to the expected value as more trials are performed.

In the context of flipping a fair coin, the expected value (theoretical probability) of getting heads on any single flip is 50%, since there are only two possible outcomes: heads or tails, each equally likely if the coin is fair. So, if you flip a coin once, you cannot predict with certainty whether it will come up heads or tails. However, as you flip the coin more and more times, the proportion of heads you observe is likely to get closer to 50%.

It's important to understand that the Law of Large Numbers speaks about "approximations" and not "certainties." In the case of 10,000 coin tosses, while you would expect the number of heads to be close to 5,000, it is possible that the actual number could be slightly more or less than this figure due to the random nature of each flip. This deviation from the expected 5,000 is a natural part of any random process and is described by the concept of standard deviation, which quantifies the amount of variation you might expect. The larger the number of flips, the smaller the proportionate variation should be in relation to the total number, which means with 10,000 flips, while you might not get exactly 5,000 heads, you wouldn't expect the number to be wildly off from 5,000 either (such as only getting 1,000 heads, which would be highly unlikely).