Question

1. a coin is flipped ten times, and turns up heads six times. a) based on these results, what is the empirical (experimental) probability of flipping heads with this coin? b) can you conclude that this coin is unfair? explain. c) explain how you can draw a better conclusion regarding the fairness of this coin.

Explanation:

Step1: Calculate empirical probability

The empirical probability $P$ of an event is given by the number of times the event occurs $n$ divided by the total number of trials $N$. Here, $n = 6$ (number of heads) and $N=10$ (number of flips). So $P=\frac{n}{N}=\frac{6}{10} = 0.6$.

Step2: Analyze fairness conclusion

A small - number of trials (only 10 flips) is not sufficient to conclude that the coin is unfair. In a fair coin, the theoretical probability of getting heads is 0.5, but due to randomness in a small number of trials, we can expect some deviation from this value.

Step3: Suggest better conclusion method

To draw a better conclusion about the fairness of the coin, we should increase the number of trials. As the number of trials approaches infinity, the empirical probability will approach the theoretical probability if the coin is fair.

Answer:

a) 0.6
b) No. A small number of trials (10 flips) is not enough to conclude unfairness as randomness can cause deviation from the theoretical 0.5 probability for a fair coin.
c) Increase the number of coin - flips. As the number of trials gets larger, the empirical probability will approach the theoretical probability if the coin is fair.