Rolling a Pair of Dice You roll a pair of six-sided dice and record the sum.

(a) List all of the possible sums and determine the probability of rolling each sum.

(b) Use a technology tool to simulate rolling a pair of dice and recording the sum 100 times. Make a tally of the 100 sums and use these results to list the probability of rolling each sum.

(c) Compare the probabilities in part (a) with the probabilities in part (b). Explain any similarities or differences.

#### Solution

(a) The sample space is

```
S={ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6),
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}.
```

The random variable $X$ is sum of the number appear on the top of two dice. $X$ take the values $2,3,\cdots, 12$.

List of all possible sum is :

. | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 |

2 | 3 | 4 | 5 | 6 | 7 | 8 |

3 | 4 | 5 | 6 | 7 | 8 | 9 |

4 | 5 | 6 | 7 | 8 | 9 | 10 |

5 | 6 | 7 | 8 | 9 | 10 | 11 |

6 | 7 | 8 | 9 | 10 | 11 | 12 |

The probability distribution of $X$ is

`$P(X=2) = P((1,1))= 1/36$`

, `$P(X=3) = P((1,2),(2,1))= 2/36$`

and so on `$P(X=12)= P((6,6))=1/36$`

.

$X$ | $P(X=x)$ | $P(X=x)$ |
---|---|---|

2 | 1/36 | 0.0278 |

3 | 2/36 | 0.0556 |

4 | 3/36 | 0.0833 |

5 | 4/36 | 0.1111 |

6 | 5/36 | 0.1389 |

7 | 6/36 | 0.1667 |

8 | 5/36 | 0.1389 |

9 | 4/36 | 0.1111 |

10 | 3/36 | 0.0833 |

11 | 2/36 | 0.0556 |

12 | 1/36 | 0.0278 |

(b) Technology tool to simulate rolling a pair of dice and recording the sum 100 times.

```
two.dice <- function(){
dice <- sample(1:6, size = 2, replace = TRUE)
return(sum(dice))
}
sim<-replicate(n = 100, expr = two.dice())
table(sim)
```

```
sim
2 3 4 5 6 7 8 9 10 11 12
4 3 10 11 14 22 12 11 6 6 1
```

`table(sim)/length(sim)`

```
sim
2 3 4 5 6 7 8 9 10 11 12
0.04 0.03 0.10 0.11 0.14 0.22 0.12 0.11 0.06 0.06 0.01
```

(c) The simulated probabilities and the theoretical probabilities are very similar.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators