Question

a coin has heads on one side and tails on the other. the coin is tossed 12 times and lands heads up 4 times. which best describes what happens when the number of trials increases significantly?
the expected frequency based on the probability of the coin landing heads up gets closer to the observed frequency.
the expected frequency based on the probability of the coin landing heads up gets closer to 1.
the observed frequency of landing heads up will always be \\(\frac{1}{3}\\) of the number of tosses.
the observed frequency of landing heads up gets closer to the expected frequency based on the probability of the coin landing heads up.

Explanation:

1. First, understand the law of large numbers: as the number of trials (tosses) increases, the observed frequency of an event (landing heads) approaches the expected frequency (based on the probability of the event).
2. Analyze each option:

• Option 1: "The expected frequency based on the probability... gets closer to observed" – Expected frequency is fixed by probability (for a fair coin, probability of heads is \( \frac{1}{2} \), expected frequency for \( n \) tosses is \( n\times\frac{1}{2} \)), it doesn't change to approach observed. Eliminate.
• Option 2: "The observed frequency... will always be \( \frac{1}{3} \) of tosses" – There's no reason it's always \( \frac{1}{3} \); the coin's probability is not \( \frac{1}{3} \) (it's a two - sided coin, fair or not, but here when trials increase, we expect it to approach its true probability, not fixed at \( \frac{1}{3} \)). Eliminate.
• Option 3: "The expected frequency... gets closer to 1" – Expected frequency is based on probability. For a coin, probability of heads is at most 1 (for a two - sided coin, it's between 0 and 1, usually \( \frac{1}{2} \) for fair). Expected frequency doesn't approach 1 unless the probability is 1, which it's not for a two - sided coin. Eliminate.
• Option 4: "The observed frequency... gets closer to expected frequency" – This is the law of large numbers. As the number of trials (tosses) increases, the actual number of heads (observed frequency) will get closer to the expected number of heads (expected frequency, which is \( n\times p \), where \( n \) is number of tosses and \( p \) is probability of heads).

Answer:

The observed frequency of landing heads up gets closer to the expected frequency based on the probability of the coin landing heads up.