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
- \(\mathrm { P } ( L = 60 )\) and
- \(\mathrm { P } ( H = 60 )\).
- Without doing any further calculations, explain which girl's score has the greater standard deviation.
- Write down
- the name of the probability distribution of \(X _ { 1 }\),
- the value of \(\mathrm { E } \left( X _ { 1 } \right)\),
- the value of \(\operatorname { Var } \left( X _ { 1 } \right)\).
- Find
(A) \(\mathrm { E } ( L )\),
(B) \(\operatorname { Var } ( L )\),
(C) \(\mathrm { E } ( H )\),
(D) \(\operatorname { Var } ( H )\).
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).
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