A game consists of spinning arrow which comes to rest pointing at one of the numbers

    \[\mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5},\text{ }\mathbf{6},\text{ }\mathbf{7},\text{ }\mathbf{8},\text{ }\mathbf{9},\text{ }\mathbf{10},\text{ }\mathbf{11},\text{ }\mathbf{12};\]

as shown below. If the outcomes are equally likely, find the probability that the pointer will point at:

    \[\left( \mathbf{i} \right)\text{ }\mathbf{6}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\]

    \[\left( \mathbf{ii} \right)\]

an even number
A game consists of spinning arrow which comes to rest pointing at one of the numbers

    \[\mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5},\text{ }\mathbf{6},\text{ }\mathbf{7},\text{ }\mathbf{8},\text{ }\mathbf{9},\text{ }\mathbf{10},\text{ }\mathbf{11},\text{ }\mathbf{12};\]

as shown below. If the outcomes are equally likely, find the probability that the pointer will point at:

    \[\left( \mathbf{i} \right)\text{ }\mathbf{6}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\]

    \[\left( \mathbf{ii} \right)\]

an even number

Solution:

We have,

Total number of possible outcomes

    \[=\text{ }12\]

    \[\left( i \right)\]

Number of favorable outcomes for

    \[6\text{ }=\text{ }1\]

6

Hence,

    \[P\left( the\text{ }pointer\text{ }will\text{ }point\text{ }at\text{ }6 \right)\text{ }=~1/12\]

    \[\left( ii \right)\]

Favorable outcomes for an even number are

    \[2,\text{ }4,\text{ }6,\text{ }8,\text{ }10,\text{ }12\]

So, number of favorable outcomes

    \[=\text{ }6\]

Hence, P(the pointer will be at an even number)

    \[=\text{ }6/12\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\]