Riddler Nation is competing against Conundrum Country at an Olympic archery event. Each team fires three arrows toward a circular target 70 meters away. Hitting the bull’s-eye earns a team 10 points, while regions successively farther away from the bull’s-eye are worth fewer and fewer points.
Whichever team has more points after three rounds wins. However, if the teams are tied after each team has taken three shots, both sides will fire another three arrows. (If they remain tied, they will continue firing three arrows each until the tie is broken.)
For every shot, each archer of Riddler Nation has a one-third chance of hitting the bull’s-eye (i.e., earning 10 points), a one-third chance of earning 9 points and a one-third chance of earning 5 points.
Meanwhile, each archer of Conundrum Country earns 8 points with every arrow.
Which team is favored to win?
Extra credit: What is the probability that the team you identified as the favorite will win?
Conundrum Country will earn 24 points in their turn. The outcomes and probability for Riddler Nation are summarized below:
| Points from 3 Rounds | Probability | Total | Result |
|---|---|---|---|
| $10,10,10$ | $\frac{1}{27}$ | $30$ | 😃 |
| $10,10,9$ | $\frac{3}{27}$ | $29$ | 😃 |
| $10,9,9$ | $\frac{3}{27}$ | $28$ | 😃 |
| $9,9,9$ | $\frac{1}{27}$ | $27$ | 😃 |
| $10,10,5$ | $\frac{3}{27}$ | $25$ | 😃 |
| $10,9,5$ | $\frac{6}{27}$ | $24$ | 😐 |
| $9,9,5$ | $\frac{3}{27}$ | $23$ | 😢 |
| $10,5,5$ | $\frac{3}{27}$ | $20$ | 😢 |
| $9,5,5$ | $\frac{3}{27}$ | $19$ | 😢 |
| $5,5,5$ | $\frac{1}{27}$ | $15$ | 😢 |
For the first 3 rounds, Riddler Nation will win $\dfrac{11}{27}$ of the time, ties will occur $\dfrac{6}{27}$ of the time, and Conundrum Country will win $\dfrac{10}{27}$ of the time.
Riddler Nation is favored to win.
Since the ratio of probabilities repeat after ties, the probability that Riddler Nation wins the event is $\dfrac{11}{11+10} = \dfrac{11}{21}$.