Question

In the 2016 Summer Olympics in Rio, there were eight runners in the final of the men's 100 meter dash. How many possible outcomes could we have seen on the podium?

(The podium honors the first three finishers in an ORDERED fashion. The first-place finishers gets the gold medal, second-place finisher gets the silver medal, and the third-place finisher gets the bronze medal.)

How many possible outcomes could we have seen on the podium that didn't include either of the two Americans? Remember the podium represents an ORDERED finish.

What is the probability that both of the American runners medalled? (To medal means to finish in the top three)

**Calculate under the assumption that order doesn't matter for this calculation.

How many possible outcomes could we have seen on the podium if we know that a runner from Jamaica finished first, a runner from America finished second, and a runner from neither Jamaica nor America finished third?

Answer 1

1) Total possible outcomes

To start with, the first place could be taken by any of the 8 finalists. (8 choices)

The second place could be taken by any of the remaining 7 finalists (7choices)

And finally, the third place can be filled by the remaining 6 contestants (6 choices)

Therefore, total number of possible outcomes = 8×7×6 = 336

2) Now, if the podium is not be filled by the 2 Americans, for the first place we have remaining 6 choices, and similarly, for 2nd and 3rd place we have 5 and 4 choices respectively.

So, total number of ways now = 6×5×4 = 120

3) Now, assuming that order doesn't matter, if we know that 2 Americans will certainly be placed on the podium (on whichever rank), the choice of just one more medalist remains. He can be any of the remaining 6 participants. So, he can be chosen in 6C1 = 6 ways.

Now, number of points in the sample space is the number of ways we can choose the medalist when order doesn't matter.

If the order doesn't matter, 3 winners from 8 can be selected in 8C3 = 56 ways.

Therefore, required probability = 6/56 = 0.107143

4) There were 2 Jamaicans and 2 Americans in the race.

The first place is taken by a Jamaican. He can be any of the two. So, we have 2 choices for this.

Similarly, an American was second, so we have 2 choices for the second place.

Now, the third place was taken by neither a Jamaican nor American. This means one of the 4 contestants from other countries took this position. So 4 choices.

Therefore, total number of ways = 2×2×4 = 16