Eight runners called A through H line up for the start of th

Eight runners (called “A” through “H”) line up for the start of the 100 meter dash. The distribution of runners’ times are all normal, with the same mean of exactly 11 seconds, but the SD’s differ. The SD’s for runners A, B, C, D, E, F, G, H are: .10, .13, .16, .19, .21, .24, .27, and .30, respectively. (A) What is the probability that A beats H? (B) Which runner has the greatest chance of winning the race? Note: for this part, you do not need to produce a rigorous proof; an intuitive argument will suffice.

Solution

A) Time difference of A and H is also normally distributed since both A and H are normal times. Also, the mean of this time difference is 0, i.e, P(time difference >0) = P( time difference < 0) = 0.5, meaning that probability that A beats H is 0.5

B) The one who wins the race typically is the one with a much lower time of finish than others. This willl be most likely for the one with highest SD, (deviation from mean), i.e, for H