Question

Which of the following statements about randomness is true?

A) A basketball player makes 80% of the shots he takes. He has made his last 10 shots, so he is due to miss his next shot because of the law of averages.
B) A fair coin is flipped 5 times and lands heads up each time. The next flip is more likely to be a tail.
C) When rolling a fair die 6 times, it is just as likely to get the sequence 123456 as it is to get 111111.
D) A spinner with three equal sections will be spun thirty times. The spinner will land in each section ten times.

Answer 1

The correct statement about randomness is option C, indicating that all sequences rolled with a fair die have equal probability. This reflects the law of large numbers in probability theory, which states that over many trials, outcomes approach the theoretical probability, despite the randomness and unpredictability of individual trials.

Step-by-step explanation:

The correct statement about randomness is: C) When rolling a fair die 6 times, it is just as likely to get the sequence 123456 as it is to get 111111. This is true because each sequence of six numbers has the same probability when rolled with a fair die.

In statistics and probability theory, the law of large numbers is an important principle. It states that as the number of experiments increases, the actual ratio of outcomes will converge on the theoretical, or expected, ratio of outcomes. For instance, while the expected theoretical probability of getting heads when tossing a fair coin is 0.5, this does not guarantee that a short series of coin tosses will result in exactly 50% heads and 50% tails. However, over a large number of tosses, the percentage of heads and tails will approach 50% each.

Games of chance are often misunderstood, leading to the gambler's fallacy, which is the mistaken belief that if something happens more frequently than normal during a given period, it will happen less frequently in the future. Option A and B are examples of this fallacy, as preceding results in random independent events do not influence future outcomes.

Randomness In Practice

For instance, a fair coin flip is always a 50-50 chance, regardless of past tosses. Similarly, each roll of a fair die is independent and has an equal chance of landing on any of the six faces. The idea of something being 'due' to happen, as in Option A, incorrectly assumes that the past events have a corrective influence on future outcomes, which is not the case in random independent events.

Lastly, the expectation that in 30 spins a three-section spinner will land on each section equally (as suggested in Option D) ignores the variability inherent in random trials. Over a very large number of spins, the distribution may approach equality, but there is no guarantee for such a small number of trials.