11 Two girls, Lili and Hui, play a game with a fair six-sided dice. The dice is thrown 10 times.
\(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) represent the scores on the \(1 ^ { \text {st } } , 2 ^ { \text {nd } } , \ldots , 10 ^ { \text {th } }\) throws of the dice.
\(L\) denotes Lili's score and \(L = 10 X _ { 1 }\).
\(H\) denotes Hui's score and \(H = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\).
- Calculate
The spreadsheet below shows a simulation of 25 plays of the game. The cell E3, highlighted, shows the score when the dice is thrown the fourth time in the first game.
\begin{table}[h]
| A | B | C | D | E | F | G | H | I | J | K | L | M | N |
| 1 | Throw of dice | Lili's | Hui's |
| 2 | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | | score | score |
| 3 | Game 1 | 3 | 5 | 2 | 1 | 1 | 3 | 1 | 1 | 1 | 4 | | 30 | 22 |
| 4 | Game 2 | 6 | 3 | 2 | 4 | 4 | 3 | 5 | 3 | 3 | 5 | | 60 | 38 |
| 5 | Game 3 | 6 | 4 | 2 | 6 | 5 | 2 | 1 | 5 | 2 | 3 | | 60 | 36 |
| 6 | Game 4 | 1 | 5 | 1 | 6 | 6 | 3 | 1 | 4 | 6 | 2 | | 10 | 35 |
| 7 | Game 5 | 4 | 4 | 3 | 1 | 6 | 4 | 4 | 1 | 6 | 2 | | 40 | 35 |
| 8 | Game 6 | 2 | 1 | 5 | 1 | 2 | 5 | 1 | 5 | 2 | 3 | | 20 | 27 |
| 9 | Game 7 | 1 | 1 | 3 | 4 | 4 | 5 | 6 | 3 | 4 | 2 | | 10 | 33 |
| 10 | Game 8 | 1 | 1 | 3 | 6 | 3 | 4 | 4 | 5 | 2 | 3 | | 10 | 32 |
| 11 | Game 9 | 2 | 2 | 2 | 4 | 3 | 2 | 1 | 5 | 5 | 6 | | 20 | 32 |
| 12 | Game 10 | 3 | 5 | 3 | 3 | 5 | 3 | 4 | 3 | 1 | 1 | | 30 | 31 |
| 13 | Game 11 | 5 | 3 | 6 | 5 | 5 | 4 | 2 | 1 | 1 | 5 | | 50 | 37 |
| 14 | Game 12 | 6 | 4 | 3 | 2 | 4 | 1 | 3 | 3 | 5 | 3 | | 60 | 34 |
| 15 | Game 13 | 2 | 3 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | | 20 | 19 |
| 16 | Game 14 | 4 | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 1 | 3 | | 40 | 30 |
| 17 | Game 15 | 5 | 1 | 2 | 6 | 3 | 4 | 6 | 3 | 6 | 4 | | 50 | 40 |
| 18 | Game 16 | 3 | 6 | 1 | 1 | 5 | 3 | 1 | 3 | 3 | 3 | | 30 | 29 |
| 19 | Game 17 | 5 | 2 | 5 | 2 | 4 | 5 | 2 | 2 | 3 | 4 | | 50 | 34 |
| 20 | Game 18 | 3 | 6 | 3 | 5 | 5 | 2 | 3 | 1 | 1 | 2 | | 30 | 31 |
| 21 | Game 19 | 6 | 6 | 3 | 1 | 5 | 6 | 3 | 4 | 1 | 6 | | 60 | 41 |
| 22 | Game 20 | 2 | 6 | 4 | 5 | 6 | 5 | 2 | 4 | 3 | 3 | | 20 | 40 |
| 23 | Game 21 | 5 | 3 | 5 | 4 | 5 | 3 | 3 | 6 | 6 | 1 | | 50 | 41 |
| 24 | Game 22 | 6 | 3 | 5 | 5 | 6 | 3 | 5 | 6 | 1 | 1 | | 60 | 41 |
| 25 | Game 23 | 5 | 4 | 5 | 5 | 6 | 4 | 2 | 1 | 3 | 6 | | 50 | 41 |
| 26 | Game 24 | 3 | 5 | 2 | 3 | 2 | 4 | 3 | 2 | 3 | 3 | | 30 | 30 |
| 27 | Game 25 | 5 | 2 | 4 | 2 | 4 | 5 | 2 | 2 | 5 | 2 | | 50 | 33 |
| 28 | | | | | | | | | | | | | | |
| 29 | | | | | | | | | | | | mean | 37.60 | 33.68 |
| 30 | | | | | | | | | | | | sd | 17.39 | 5.77 |
\captionsetup{labelformat=empty}
\caption{Fig. 11}
- Use the simulation to estimate \(\mathrm { P } ( L > 40 )\) and \(\mathrm { P } ( H > 40 )\).
- (A) Calculate the exact value of \(\mathrm { P } ( L > 40 )\).
(B) Comment on how the exact value compares with your estimate of \(\mathrm { P } ( L > 40 )\) in part (v).
Hui wonders whether it is appropriate to use the Central Limit Theorem to approximate the distribution of \(X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\). - (A) State what type of diagram Hui could draw, based on the output from the spreadsheet, to investigate this.
(B) Explain how she should interpret the diagram. - (A) Calculate an approximate value of \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 } > 40 \right)\) using the Central Limit Theorem.
(B) Comment on how this value compares with your estimate of \(\mathrm { P } ( H > 40 )\) in part (v).
\section*{Copyright Information:}
}{www.ocr.org.uk}) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the