Question

Run the following experiment 20 times: (a) Flip 5 coins. (b) Record the number of heads. (c) Repeat (a)-(b) until you have 20 data points. (d) What proportion (fraction) of your experiments land on heads three times? Call this calculation pˆ. (e) Theoretically, your proportion should be the same as the probability of success. However, it is not. Explain why.

Answer 1

The probability of success : (a) in flipping three heads in five coin flips is 0.3125, with a failure probability of 0.6875 (b). In 20 experiments (c), the observed proportion (p-hat) of landing on three heads is 0.3 (d), differing slightly from the theoretical probability. This discrepancy is due to random variations in finite trials (e), as the law of large numbers predicts convergence with increased trials.

(a) The probability of success (flipping three total heads in five coin flips) can be calculated using the binomial probability formula:

p = 10 * 0.125 * 0.25

p = 0.3125

So, the probability of success (p) is 0.3125.

(b) The probability of failure (q) is complementary to the probability of success and can be calculated as:

q = 1 - p = 1 - 0.3125 = 0.6875

So, the probability of failure (q) is 0.6875.

(c) In each of the 20 experiments, flip 5 coins, record the number of heads, and repeat until you have 20 data points.

(d) Calculate the proportion (p-hat) of experiments that land on three heads. This is the fraction of the 20 experiments where exactly three heads are obtained.

Let's say in one experiment, the outcomes are as follows:

Experiment 1: 3 heads, 2 tails

In another experiment, you might get:

Experiment 2: 2 heads, 3 tails

Repeat this process for all 20 experiments and count how many times you get exactly three heads. Let's assume, for example, that in 6 out of the 20 experiments, you get three heads.

`\[ p-hat = \frac{\text{Number of experiments with exactly three heads}}{\text{Total number of experiments}} \]`

`\[ p-hat = (6)/(20) \]`

`\[ p-hat = 0.3 \]`

So, the observed proportion (p-hat) is 0.3.

(e) Theoretically, the proportion (p) should be the same as the probability of success. However, in practice, p-hat might differ from p due to random variations inherent in finite experiments. The law of large numbers suggests that as the number of trials increases, p-hat should converge to p, but with a limited number of trials, there can be discrepancies.