Problem C and Solution

Just Sum Dice

Ahmik created a game for his schoolâ€™s carnival where players roll two dice and find the sum of the two numbers on the top faces. If this sum is a perfect square or a prime number, they win a prize. To make it more interesting, Ahmik made the two dice using a 3D printer so that they each have the numbers \(1\), \(2\), \(3\), \(5\), \(7\), and \(9\) on their faces. One of the dice is purple and the other is green.

What is the probability that a player will win a prize after rolling the dice once?

Note:

A square of any integer is called a *perfect square*. The number \(25\) is a perfect square since it can be expressed as \(5^2\) or \(5 \times 5\).

A *prime number* is an integer greater than \(1\) that has only two positive divisors; \(1\) and itself. The number \(17\) is prime because its only positive divisors are \(1\) and \(17\).

To solve this problem, we will create a table showing all of the possible rolls for each die and the corresponding sums.

Green Die | |||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 5 | 7 | 9 | ||

Purple Die | 1 | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(10\) |

2 | \(3\) | \(4\) | \(5\) | \(7\) | \(9\) | \(11\) | |

3 | \(4\) | \(5\) | \(6\) | \(8\) | \(10\) | \(12\) | |

5 | \(6\) | \(7\) | \(8\) | \(10\) | \(12\) | \(14\) | |

7 | \(8\) | \(9\) | \(10\) | \(12\) | \(14\) | \(16\) | |

9 | \(10\) | \(11\) | \(12\) | \(14\) | \(16\) | \(18\) |

From the table, we see that there are \(36\) possible outcomes. We also see that the perfect squares \(4\), \(9\), and \(16\) appear in the table seven times.

The smallest number in the table is \(2\), and the largest number in the table is \(18\). The prime numbers appearing in the table in this range of numbers are \(2\), \(3\), \(5\), \(7\), and \(11\). These numbers appear in the table a total of nine times.

Thus there are \(7\) perfect squares and \(9\) prime numbers in the table. Since a number cannot be both a prime number and a perfect square, we can conclude that there are \(7+9=16\) sums that are prime numbers or perfect squares.

To determine the probability of a specific outcome, we divide the number of times the specific outcome occurs by the total number of possible outcomes. Thus, the probability of a player rolling a sum that is either a prime number or a perfect square is \(16\div 36=\frac{4}{9}\approx 44\%\). Therefore, a player has approximately a \(44\%\) chance of winning a prize after rolling the dice once.

**Extension:** A game is considered fair if the chance of winning is \(50\%\). How could you change the rules of this game to make it fair?