Question

3.3 four games, one winner. below are four versions of the same game. your arch - nemesis gets to pick the version of the game, and then you get to choose how many times to flip a coin: 10 times or 100 times. it costs $1 to play each game. (a) if the proportion of heads is larger than 0.60, you win $1. (b) if the proportion of heads is larger than 0.40, you win $1. (c) if the proportion of heads is between 0.40 and 0.60, you win $1. (d) if the proportion of heads is smaller than 0.30, you win $1. 3.3 (a) 10 tosses. fewer tosses mean more variability in the sample fraction of heads, meaning theres a better chance of getting at least 60% heads. (b) 100 tosses. more flips means the observed proportion of heads would often be closer to the average, 0.50. (c) 100 tosses. with more flips, the observed proportion of heads would often be closer to the average, 0.50. (d) 10 tosses. fewer flips would increase variability in the fraction of tosses that are heads.

Explanation:

Step1: Recall law of large numbers

The law of large numbers states that as the number of trials (coin - flips here) increases, the sample proportion of heads gets closer to the true proportion (0.5 for a fair coin).

Step2: Analyze option (a)

For 10 flips, there is a relatively high chance of getting a proportion far from 0.5. The probability of getting a proportion larger than 0.6 is not extremely low due to the small number of trials.

Step3: Analyze option (b)

With 100 flips, due to the law of large numbers, the sample proportion is more likely to be close to 0.5. The probability of getting a proportion larger than 0.4 is high as the distribution of sample proportions will be more concentrated around 0.5 for 100 trials compared to 10 trials.

Step4: Analyze option (c)

For 100 flips, getting a proportion between 0.4 and 0.6 is quite likely because as the number of trials increases, the sample proportions cluster more tightly around 0.5.

Step5: Analyze option (d)

For 10 flips, getting a proportion smaller than 0.3 is more possible compared to 100 flips since there is more variability in a smaller number of trials.

To maximize the chance of winning:

• For option (a) (proportion of heads larger than 0.6), choose 10 flips because with fewer trials, there is more chance of getting an extreme proportion.
• For option (b) (proportion of heads larger than 0.4), choose 100 flips as the proportion is more likely to be above 0.4 with more trials.
• For option (c) (proportion of heads between 0.4 and 0.6), choose 100 flips as the sample proportion is more likely to be in this range with more trials.
• For option (d) (proportion of heads smaller than 0.3), choose 10 flips as there is more variability and a higher chance of getting an extreme low - proportion with fewer trials.

Answer:

(a) 10 flips
(b) 100 flips
(c) 100 flips
(d) 10 flips