Betty and Jane are gambling. They are cutting cards (picking a random place in the deck to see a card). Whichever one has the higher card wins the bet. If the cards have the same value, they try again. Betty and Jane do this 100 times. Tom and Bill are doing the same thing but only betting 10 times. Is it Bill or Betty who is more likely to end up having very close to 50% wins?

Question

Betty and Jane are gambling. They are cutting cards​ (picking a random place in the deck to see a​ card). Whichever one has the higher card wins the bet. If the cards have the same​ value, they try again. Betty and Jane do this 100 times. Tom and Bill are doing the same thing but only betting 10 times. Is it Bill or Betty who is more likely to end up having very close to​ 50% wins?

BlackTom AI · Accepted Answer

First, the question sets up two gamblers, Betty and Jane with 100 trials, and Bill and Tom with 10 trials, all performing the same kind of random card-cutting experiment where the higher card wins and ties are re-played. The central concept to evaluate is how the Law of Large Numbers (LLN) affects the proximity of observed win rates to the true probability as the number of trials changes.
Option 1: 'They are both equally likely to be close to 50% wins according to the Law of Large Numbers.' This statement is not correct as written because the LLN does not say both will be equally likely to be near 50% just by adjusting the number of trials; it says that as the number of trials grows, the observed proportion converges to the true probability. Since Betty conducts 100 trials and Bill conducts 10, the distributions of their win proportions around the true probability behave differently. So this option ignores the impact of sample size on the variability of the observed proportion.
Option 2: 'Betty is more likely to end up having close to 50% wins as she is betting more times and the Law of Large Numbers says that the more times a random experiment is repeated the closer it comes to the true probability.' This option correctly cites the LLN about convergence to the true probability with more repetitions, but it misuses the implication: more trials generally reduce variability, making it easier for the observed win rate to be near the true probability. However, being closer to 50% specifically depends on the true probability of winning each round. If the true probability is near 0.5, both sequences would tend toward 0.5, but more trials make the proportion more stable around that value rather than necessarily closer to 50% more than fewer trials. The statement could be read as overly simplistic and potentially misleading about which person is more likely to be near 50% just because they have more trials.
Option 3: 'Bill is more likely to end up having close to 50% wins as Betty is betting more times than him so it is unlikely she will be close to 50% wins.' This is conflating sample size with proximity to 50% in a way that isn’t strictly correct. While Bill’s 10 trials produce a wider spread of possible win rates, that does not guarantee that Betty, with 100 trials, will be unable to be close to 50%. In fact, with 100 trials, if the true win probability per trial is 0.5, Betty’s observed proportion is likely to be very close to 0.5, while Bill’s could deviate more. The reasoning here incorrectly asserts that Betty cannot be close to 50% due to having more trials.
Option 4: 'Bill is more likely to end up having close to 50% wins as he is only betting 10 times and it is possible he wins exactly 5 times while Betty is betting more times so it is unlikely she will win exactly 50 times in accordance with the Law of Large Numbers.' This option correctly identifies that with fewer trials (Bill, 10), the observed win rate has higher variability and can be exactly 50% more plausibly (e.g., 5 wins out of 10). However, the claim that Betty is unlikely to be exactly 50% because she bets more times is not precisely a direct consequence of LLN; LLN says variance decreases with more trials, but it does not guarantee exact 50% is impossible for Betty. The key idea remains that smaller sample sizes (Bill) can land exactly at 50% more often than larger sample sizes, due to greater relative variability. This option captures the practical effect of sample size on the closeness to 50% better than the others, though it couches it with a specific LLN phrasing.
In summary, the crux is that smaller sample sizes exhibit greater randomness, making exact 50% outcomes more plausible, while larger samples produce a more stable but possibly different reality depending on the true probability. The most consistent choice with this understanding is that the person with the smaller number of trials (Bill) is more likely to be near 50% simply due to higher sampling variability, whereas Betty with many trials will tend to stabilize near the true probability, which is around 0.5 if the cards are symmetric. The nuanced caveat is that none of the options are perfectly worded, but the one emphasizing Bill’s higher chance of exactly 5 wins aligns best with the standard intuition from sampling variability and LLN.

类似问题

Select all statements that are correct. The law of large numbers and central limit theorem (taken together) imply that:

As the sample size increases, the expected value of noise Stays the Same

Which of the following statements is true about the Law of Large Numbers?

Consider two pairs of grandparents. The first pair has 4 grandchildren and the second pair has 31 grandchildren. Which of the two pairs is more likely to have between 40% and 60% boys as grandchildren, assuming that boys and girls are equally likely as children? Why?

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