Question

The two dice are tossed. What is the probably of the event that the sum of two numbers on the both dice at least equal 6 and at most equal 9?

Answer 1

`P_("["6,9"]")=5/9`

Explanation:

With out loss of generality we can assume one die is `color(red)("red")` and the second die is `color(green)("green")`

For each of the `color(red)(6)` faces on the `color(red)("red die")` there are color(green)(6) different possible outcomes on the `color(green)("green die")`.
`rArr` there are `color(red)(6) xx color(green)(6) = color(blue)(36)` possible combined outcomes.

Of these outcomes

A total of 6 can be achieved in `color(cyan)(5)` ways:`{(color(red)(1),color(green)(5)),(color(red)(2),color(green)(4)),(color(red)(3),color(green)(3)),(color(red)(4),color(green)(2)),(color(red)(5),color(green)(1))}`

A total of 7 can be achieved in `color(cyan)(6)` ways:`{(color(red)(1),color(green)(6)),(color(red)(2),color(green)(5)),(color(red)(3),color(green)(4)),(color(red)(4),color(green)(3)),(color(red)(5),color(green)(2)),(color(red)(6),color(green)(1))}`

A total of 8 can be achieved in `color(cyan)(5)` ways:`{(color(red)(2),color(green)(6)),(color(red)(3),color(green)(5)),(color(red)(4),color(green)(4)),(color(red)(5),color(green)(3)),(color(red)(6),color(green)(2))}`

A total of 9 can be achieved in `color(cyan)(4)` ways:`{(color(red)(3),color(green)(6)),(color(red)(4),color(green)(5)),(color(red)(5),color(green)(4)),(color(red)(6),color(green)(3))}`

Since these event are mutually exclusive there are
`color(white)("XXX")color(cyan)(5+6+5+4) =color(brown)(20)` ways of achieving `{6,7,8,9}`

So the probability of achieving `in{6,7,8,9}`is
`color(white)("XXX")color(brown)(20)/color(blue)(36)= 4/9`