2 Leila and Caleb are playing a game, using fair six-sided dice and unbiased coins.
- Leila rolls two dice, and her score \(L\) is the total of the scores on the two dice.
- Caleb spins 4 coins and his score \(C\) is three times the number of heads obtained.
The winner of a game is the player with the higher score. If the two scores are equal, the result of the game is a draw. The spreadsheet in Fig. 2 shows a simulation of 20 plays of the game.
\begin{table}[h]
| A | B | C | D | E | F | G | H |
| 1 | First dice | Second dice | Total (Leila's score) \(\boldsymbol { L }\) | Coin 1 | Coin 2 | Coin 3 | Coin 4 | Caleb's score \(\boldsymbol { C }\) |
| 2 | 1 | 2 | 3 | H | T | T | T | 3 |
| 3 | 6 | 1 | 7 | T | H | T | T | 3 |
| 4 | 2 | 6 | 8 | H | H | T | T | 6 |
| 5 | 2 | 5 | 7 | T | H | H | H | 9 |
| 6 | 1 | 5 | 6 | T | H | T | T | 3 |
| 7 | 5 | 2 | 7 | H | H | H | H | 12 |
| 8 | 1 | 1 | 2 | H | T | H | T | 6 |
| 9 | 2 | 6 | 8 | T | H | T | H | 6 |
| 10 | 6 | 2 | 8 | H | T | H | T | 6 |
| 11 | 1 | 3 | 4 | T | H | H | H | 9 |
| 12 | 6 | 1 | 7 | T | H | T | T | 3 |
| 13 | 3 | 1 | 4 | T | T | T | T | 0 |
| 14 | 3 | 6 | 9 | H | T | H | H | 9 |
| 15 | 2 | 3 | 5 | T | H | H | H | 9 |
| 16 | 2 | 5 | 7 | H | H | H | H | 12 |
| 17 | 1 | 5 | 6 | H | H | T | H | 9 |
| 18 | 5 | 6 | 11 | T | H | H | H | 9 |
| 19 | 4 | 2 | 6 | T | H | H | T | 6 |
| 20 | 6 | 5 | 11 | T | T | H | H | 6 |
| 21 | 1 | 1 | 2 | T | T | T | T | 0 |
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{table}
- Explain why the value of \(C\) in row 2 is 3 .
- Use the spreadsheet to estimate \(\mathrm { P } ( C > 6 )\) and \(\mathrm { P } ( L > 6 )\).
- Use the spreadsheet to estimate the probability that Leila loses a randomly chosen game.
- Explain why your answers to parts (b) and (c) may not be very close to the true values.
- Leila claims that the game is fair (that Leila and Caleb each have an equal chance of winning) because both she and Caleb can get a maximum score of 12 and also in the simulation she won exactly \(50 \%\) of the games.
Make 2 comments about Leila’s claim.