Question

1. what is the least number of rolls it will take to release all your prisoners?
2. what is the probability of having the difference in dice be 0?
3. what are the odds of the difference being 3?
4. what is the probability of rolling a difference of 4?

Explanation:

Step1: Calculate total number of outcomes

When rolling two - six - sided dice, the total number of possible outcomes is \(n(S)=6\times6 = 36\).

Step2: Solve for the probability of difference being 0

The pairs where the difference between the two dice is 0 are \((1,1)\), \((2,2)\), \((3,3)\), \((4,4)\), \((5,5)\), \((6,6)\). So \(n(0)=6\). The probability \(P(0)=\frac{n(0)}{n(S)}=\frac{6}{36}=\frac{1}{6}\).

Step3: Solve for the odds of difference being 3

The pairs where the difference is 3 are \((1,4)\), \((2,5)\), \((3,6)\), \((4,1)\), \((5,2)\), \((6,3)\). So \(n(3)=6\). The odds in favor of an event \(E\) is given by \(\frac{n(E)}{n(\text{not }E)}\). Here, \(n(\text{not }3)=36 - 6=30\). The odds of the difference being 3 is \(\frac{6}{30}=\frac{1}{5}\).

Step4: Solve for the probability of difference being 4

The pairs where the difference is 4 are \((1,5)\), \((2,6)\), \((5,1)\), \((6,2)\). So \(n(4)=4\). The probability \(P(4)=\frac{n(4)}{n(S)}=\frac{4}{36}=\frac{1}{9}\).

Step5: Analyze the first question (least number of rolls to release all prisoners)

Since there is no information about the relationship between dice - rolls and releasing prisoners, we cannot answer this question.

Answer:

1. \(\frac{1}{6}\)
2. \(\frac{1}{5}\)
3. \(\frac{1}{9}\)
4. Cannot be answered due to lack of information.