If you flip a fair coin, the probability of getting a heads is $$ 0.50=50 \% $$. Which of the following statements best describes the Law of Large Numbers as applied to the experimental proportion (we called this experimental probability in class) of flipping heads after $$ n $$ flips? As the number of flips n increases, the experimental proportion of heads approaches $$ 50 \% $$ but will never equal $$ 50 \% $$. The experimental proportion of heads will start below $$ 50 \% $$ and will get closer to $$ 50 \% $$ as $$ n $$ increases O Eventually, after some number n of flips, the experimental proportion of heads will equal $$ 50 \% $$ and remain $$ 50 \% $$. O As the number of flips n increases, the experimental proportion of heads approaches $$ 50 \% $$, though might never perfectly equal $$ 50 \% $$. A

Question

UpStudy AI · Accepted Answer

The Law of Large Numbers states that as the number of trials (in this case, coin flips) increases, the experimental probability (or experimental proportion) of an event (getting heads) will converge to the theoretical probability of that event.

In the context of flipping a fair coin, the theoretical probability of getting heads is $$ 50\% $$ or $$ 0.50 $$. As you flip the coin more times, the proportion of heads observed in those flips will get closer and closer to $$ 0.50 $$. However, due to the nature of random events, it is possible that the experimental proportion may never perfectly equal $$ 0.50 $$ at any finite number of flips, but it will get arbitrarily close.

Now, let's analyze the provided statements:

1. **As the number of flips $$ n $$ increases, the experimental proportion of heads approaches $$ 50\% $$ but will never equal $$ 50\% $$.**
   - This statement is misleading because while it is true that the proportion approaches $$ 50\% $$, it is not guaranteed that it will never equal $$ 50\% $$ at some point.

2. **The experimental proportion of heads will start below $$ 50\% $$ and will get closer to $$ 50\% $$ as $$ n $$ increases.**
   - This statement is not accurate because the experimental proportion can start above, below, or exactly at $$ 50\% $$ depending on the outcomes of the initial flips.

3. **Eventually, after some number $$ n $$ of flips, the experimental proportion of heads will equal $$ 50\% $$ and remain $$ 50\% $$.**
   - This statement is incorrect because it implies that the proportion will stabilize at $$ 50\% $$, which is not guaranteed.

4. **As the number of flips $$ n $$ increases, the experimental proportion of heads approaches $$ 50\% $$, though might never perfectly equal $$ 50\% $$.**
   - This statement accurately reflects the Law of Large Numbers. It acknowledges that while the experimental proportion will get closer to $$ 50\% $$, it may not necessarily equal $$ 50\% $$ at any finite number of flips.

Based on this analysis, the best description of the Law of Large Numbers as applied to the experimental proportion of flipping heads after $$ n $$ flips is:

**As the number of flips $$ n $$ increases, the experimental proportion of heads approaches $$ 50\% $$, though might never perfectly equal $$ 50\% $$.**
As you flip a coin more times, the number of heads you get will get closer to half of the total flips, but it might never be exactly 50% in any single set of flips.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Probability Questions