Question

4. A drawer contains 21 black socks and 9 white socks. A number generator simulates randomly selecting 10 socks from the drawer. The number generator is used 10 times and the number of black socks in each trial is shown in the plot.
Does this number generator appear to be fair?

WHY OR WHY NOT?


I know the answer is yes it appears to be fair, but please answer WHY. Thank you :)

Answer 1

There are 21 black socks and 9 white socks. Theoretically, the probability of picking a black sock is 21/(21+9) = 21/30 = 0.70 = 70%

Assuming we select any given sock, and then put it back (or replace it with an identical copy), then we should expect about 0.70*10 = 7 black socks out of the 10 we pick from the drawer. If no replacement is made, then the expected sock count will likely be different.

The dot plot shows the data set is

{5, 5, 6, 6, 7, 7, 7, 8, 8, 8}

The middle-most value is between the first two '7's, so the median is (7+7)/2 = 14/2 = 7. This can be thought of as the average expected number of black socks to get based on this simulation. So that's why I consider it a fair number generator because it matches fairly closely with the theoretical expected number of black socks we should get. Again, this is all based on us replacing each sock after a selection is made.