1 Abby runs a stall at a charity event. Visitors to the stall pay to play a game in which six fair dice are rolled. If the difference between the highest and lowest scores is less than 3 then the player wins \(\pounds 5\). Otherwise the player wins nothing.
Abby designs the spreadsheet shown in Fig. 1 to estimate the probability of a player winning, by simulating 20 goes at the game. Cell C5, highlighted, shows that the 2nd dice in simulated game 4 scores 5 . Cells H5 and I5 show the highest and lowest scores, respectively, in game 4, and cell J5 gives the difference between them.
\begin{table}[h]
| C5 | \(\times \vee f _ { x }\) | =RANDBETWEEN(1,6) |
| A | B | C | D | E | F | G | H | | J |
| 1 | | dice 1 | dice 2 | dice 3 | dice 4 | dice 5 | dice 6 | High score | Low score | Difference |
| 2 | game 1 | 2 | 2 | 4 | 2 | 3 | 3 | 4 | 2 | 2 |
| 3 | game 2 | 2 | 6 | 3 | 2 | 1 | 2 | 6 | 1 | 5 |
| 4 | game 3 | 3 | 1 | 5 | 3 | 4 | 6 | 6 | 1 | 5 |
| 5 | game 4 | 6 | 5 | 2 | 5 | 6 | 3 | 6 | 2 | 4 |
| 6 | game 5 | 6 | 3 | 3 | 5 | 3 | 2 | 6 | 2 | 4 |
| 7 | game 6 | 5 | 6 | 3 | 5 | 1 | 4 | 6 | 1 | 5 |
| 8 | game 7 | 2 | 3 | 1 | 2 | 6 | 4 | 6 | 1 | 5 |
| 9 | game 8 | 6 | 6 | 6 | 6 | 1 | 5 | 6 | 1 | 5 |
| 10 | game 9 | 3 | 6 | 2 | 5 | 4 | 1 | 6 | 1 | 5 |
| 11 | game 10 | 5 | 1 | 1 | 4 | 6 | 1 | 6 | 1 | 5 |
| 12 | game 11 | 2 | 5 | 6 | 1 | 6 | 5 | 6 | 1 | 5 |
| 13 | game 12 | 2 | 5 | 6 | 6 | 6 | 6 | 6 | 2 | 4 |
| 14 | game 13 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 |
| 15 | game 14 | 1 | 6 | 6 | 6 | 3 | 5 | 6 | 1 | 5 |
| 16 | game 15 | 2 | 2 | 3 | 3 | 5 | 1 | 5 | 1 | 4 |
| 17 | game 16 | 1 | 2 | 3 | 4 | 3 | 3 | 4 | 1 | 3 |
| 18 | game 17 | 5 | 2 | 4 | 2 | 1 | 6 | 6 | 1 | 5 |
| 19 | game 18 | 6 | 1 | 5 | 2 | 1 | 5 | 6 | 1 | 5 |
| 20 | game 19 | 1 | 3 | 5 | 1 | 3 | 5 | 5 | 1 | 4 |
| 21 | game 20 | 5 | 4 | 3 | 2 | 5 | 1 | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{table}
- (A) Write down the numbers in columns H , I and J for game 20 .
(B) Use the spreadsheet to estimate the probability of a player winning a game. - State how the estimate of probability in (i) (B) could be improved.
- Give one advantage and one disadvantage of using this simulation technique compared with working out the theoretical probability.
All profit made by the stall is given to charity. Abby has to decide how much to charge players to play.
- If Abby charges \(\pounds 1\) per game, estimate the total profit when 50 players each play the game once.