You are interested in the number of chocolate chips on a cookie. To estimate this number, you consider two different experiments:
A. Pick a cookie at random, all cookies being equally likely to be picked, and count the number of chocolate chips on that cookie.
B. Pick a chocolate chip at random, all chocolate chips being equally likely to be picked, and count the number of chocolate chips on the cookie to which the selected chocolate chip belongs.
Which method will tend to give higher results?
b
Method B will tend to give higher results.
In Method A, when picking a cookie at random, the number of chocolate chips on that cookie could vary widely. Some cookies might have very few chocolate chips, while others might have a large number. So, the average number of chocolate chips over a large number of cookies will be influenced by the variability in the number of chocolate chips on each cookie.
However, in Method B, when picking a chocolate chip at random, the number of chocolate chips on the cookie to which the selected chip belongs will always be at least 1, since we selected a chocolate chip in the first place. Therefore, the average number of chocolate chips over a large number of randomly selected chips will tend to be higher compared to the average number of chocolate chips over a large number of cookies.
To determine which method will tend to give higher results, we need to consider the underlying probabilities and distributions of each method.
Method A involves randomly picking a cookie and counting the number of chocolate chips on that specific cookie. The number of chocolate chips on each cookie is variable, and the probability distribution of the number of chocolate chips may vary depending on the specific batch of cookies. Since we are picking cookies randomly, each cookie has an equal chance of being selected. Therefore, the number of chocolate chips on the randomly selected cookie will follow the distribution of the number of chocolate chips across all cookies.
Method B involves randomly picking a chocolate chip and then counting the number of chocolate chips on the cookie that the selected chocolate chip belongs to. Since we are picking a chocolate chip randomly, and assuming all chocolate chips are equally distributed across the cookies, each chocolate chip has an equal chance of being selected. Therefore, the number of chocolate chips on the cookie that the selected chocolate chip belongs to will also follow the distribution of the number of chocolate chips across all cookies.
In both methods, the number of chocolate chips on the cookies and the number of chocolate chips on the selected cookie are following the same underlying distribution. Therefore, the expected or average number of chocolate chips would be the same for both methods.
However, if you are looking for the method that tends to give higher results, Method B is more likely to do so. This is because as you randomly select a chocolate chip from a cookie, you are more likely to select a cookie with a higher number of chocolate chips, since cookies with more chocolate chips will have a higher chance of having their chocolate chips selected. This can potentially lead to higher estimates of the number of chocolate chips compared to randomly selecting a cookie in Method A.
So, in summary, Method B, which involves picking a chocolate chip at random and then counting the number of chocolate chips on the cookie it belongs to, tends to give higher results compared to Method A, which involves randomly picking a cookie and counting the number of chocolate chips on that specific cookie.