Question

question 4
consider the roll of 2 fair 6 - sided dice.
a) are the outcomes of the rolls dependent or independent?
enter answer
b) what is the probability of getting a sum of 7? enter your answer as a fraction using the / key.
enter answer
c) what is the probability of getting a sum of 11? enter your answer as a fraction using the / key.
enter answer
d) what is the probability of getting a sum of 7 or 11? enter your answer as a fraction using the / key.
enter answer
12 points

Explanation:

Step1: Determine dependence of outcomes

When rolling two fair 6 - sided dice, the outcome of one die does not affect the outcome of the other die. For example, if the first die shows a 3, the second die still has 6 possible outcomes (1, 2, 3, 4, 5, 6) regardless. So, the outcomes of the rolls are independent.

Step2: Calculate total number of outcomes

By the fundamental counting principle, if die 1 has \(n_1 = 6\) outcomes and die 2 has \(n_2=6\) outcomes, the total number of outcomes when rolling two dice is \(n=n_1\times n_2=6\times6 = 36\)

Step3: Find number of ways to get a sum of 7

Let the outcomes of the two dice be \((x,y)\) where \(x + y=7\) and \(1\leq x\leq6\), \(1\leq y\leq6\). The pairs \((x,y)\) are \((1,6)\), \((2,5)\), \((3,4)\), \((4,3)\), \((5,2)\), \((6,1)\). So, \(m_7 = 6\)
The probability \(P(\text{sum}=7)=\frac{m_7}{n}=\frac{6}{36}=\frac{1}{6}\)

Step4: Find number of ways to get a sum of 11

Let \(x + y = 11\) with \(1\leq x\leq6\), \(1\leq y\leq6\). The pairs \((x,y)\) are \((5,6)\), \((6,5)\). So, \(m_{11}=2\)
The probability \(P(\text{sum}=11)=\frac{m_{11}}{n}=\frac{2}{36}=\frac{1}{18}\)

Step5: Calculate probability of sum of 7 or 11

Using the formula \(P(A\cup B)=P(A)+P(B)\) (since \(A\) (sum = 7) and \(B\) (sum = 11) are mutually - exclusive events, i.e., \(A\cap B=\varnothing\)).
\(P(\text{sum}=7\text{ or sum}=11)=\frac{6 + 2}{36}=\frac{8}{36}=\frac{2}{9}\)

Answer:

a) Independent
b) \(\frac{1}{6}\)
c) \(\frac{1}{18}\)
d) \(\frac{2}{9}\)