Question

Two dice are thrown. Describe the sample space of this experiment.

Answer 1

Hint:
1) If two dice are thrown, there are 6 × 6 = 36 different outcomes possible.
2) The sample space of a random experiment is the set of all possible outcomes.
3) The sample space is represented using S.
4) A subset of the sample space of an experiment is called an event represented by E.

Complete step by step solution:
When two dice are thrown, we may get an outcome as (1, 1), (2, 5), (1, 6), (3, 1) etc.
Since, there are six different possible outcomes for a dice, the set (S) of all the outcomes can be listed as follows:
\[\left( {1,{\text{ }}1} \right),{\text{ }}\left( {1,{\text{ }}2} \right),{\text{ }}\left( {1,{\text{ }}3} \right),{\text{ }}\left( {1,{\text{ }}4} \right),{\text{ }}\left( {1,{\text{ }}5} \right),{\text{ }}\left( {1,{\text{ }}6} \right){\text{ }} = {\text{ }}6{\text{ }}possibilities.\]
\[\left( {2,{\text{ }}1} \right),{\text{ }}\left( {2,{\text{ }}2} \right),{\text{ }}\left( {2,{\text{ }}3} \right),{\text{ }}\left( {2,{\text{ }}4} \right),{\text{ }}\left( {2,{\text{ }}5} \right),{\text{ }}\left( {2,{\text{ }}6} \right){\text{ }} = {\text{ }}6{\text{ }}possibilities.\]
\[\left( {3,{\text{ }}1} \right),{\text{ }}\left( {3,{\text{ }}2} \right),{\text{ }}\left( {3,{\text{ }}3} \right),{\text{ }}\left( {3,{\text{ }}4} \right),{\text{ }}\left( {3,{\text{ }}5} \right),{\text{ }}\left( {3,{\text{ }}6} \right){\text{ }} = {\text{ }}6{\text{ }}possibilities.\]
\[\;\left( {4,{\text{ }}1} \right),{\text{ }}\left( {4,{\text{ }}2} \right),{\text{ }}\left( {4,{\text{ }}3} \right),{\text{ }}\left( {4,{\text{ }}4} \right),{\text{ }}\left( {4,{\text{ }}5} \right),{\text{ }}\left( {4,{\text{ }}6} \right){\text{ }} = {\text{ }}6{\text{ }}possibilities.\]
\[\;\left( {5,{\text{ }}1} \right),{\text{ }}\left( {5,{\text{ }}2} \right),{\text{ }}\left( {5,{\text{ }}3} \right),{\text{ }}\left( {5,{\text{ }}4} \right),{\text{ }}\left( {5,{\text{ }}5} \right),{\text{ }}\left( {5,{\text{ }}6} \right){\text{ }} = {\text{ }}6{\text{ }}possibilities.\]
\[\;\left( {6,{\text{ }}1} \right),{\text{ }}\left( {6,{\text{ }}2} \right),{\text{ }}\left( {6,{\text{ }}3} \right),{\text{ }}\left( {6,{\text{ }}4} \right),{\text{ }}\left( {6,{\text{ }}5} \right),{\text{ }}\left( {6,{\text{ }}6} \right){\text{ }} = {\text{ }}6{\text{ }}possibilities.\]
Total number of elements (possibilities) of set S are therefore,\[n\left( S \right) = 6 \times 6 = 36\]; i.e. six possibilities of second dice for each of the six possibilities of the first dice.

Note:
1) A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set.
2) The probability of an outcome E in a sample space S is a number P between 1 and 0 that measures the likelihood that E will occur on a single trial.