Question

identify the sample space of the probability experiment and determine the number of outcomes in the sample space. draw a tree diagram.
rolling two six - sided dice one after another
identify the sample space of the probability experiment.
a. {1, 2, 3, 4, 5, 6}
b. {11, 22, 33, 44, 55, 66}
c. {11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66}
d. {1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6}
there are □ outcomes in the sample space.
(type a whole number.)

Explanation:

Step1: Analyze the sample space for rolling two dice

When rolling two six - sided dice, the first die can show 6 possible outcomes (\(1,2,3,4,5,6\)), and for each outcome of the first die, the second die can also show 6 possible outcomes.
The sample space is the set of all ordered pairs \((i,j)\) where \(i\) is the outcome of the first die and \(j\) is the outcome of the second die.
For example, if the first die shows \(1\) and the second die shows \(1\), the outcome is \((1,1)\); if the first die shows \(1\) and the second die shows \(2\), the outcome is \((1,2)\), and so on.
The sample space is \(\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}\) which is equivalent to option C.

Step2: Calculate the number of outcomes

By the multiplication principle, if event \(A\) (rolling the first die) has \(n(A)=6\) outcomes and event \(B\) (rolling the second die) has \(n(B) = 6\) outcomes, then the number of outcomes of the combined event (rolling two dice) is \(n=n(A)\times n(B)\).
So, \(n = 6\times6=36\).

Answer:

C. \(\{11,12,13,14,15,16,21,22,23,24,25,26,31,32,33,34,35,36,41,42,43,44,45,46,51,52,53,54,55,56,61,62,63,64,65,66\}\)
There are \(36\) outcomes in the sample space.