Question

For the experiment of tossing a six sided die twice, what is the probability that the sum is less than 11?

Answer 1

`P("sum" < 11)=11/12`

Explanation:

Without loss of generality we could assume that one die is `color(red)("red")` and the other is `color(green)("green")`

For each of the 6 `color(red)("red")` outcomes there are 6 `color(green)("green")` outcomes.
This means that there are a combined collection of `color(red)6xxcolor(green)6=color(blue)(36)` possible outcomes.

Of the `color(blue)(36)` possible outcomes only `color(magenta)3` result in totals greater than on equal to 11; namely
`color(white)("XXX")< color(red)("red"),color(green)("green") > in { < color(red)5, color(green)6 >, < color(red)6, color(green)5 >, < color(red)6, color(green)6 >}`

Therefore, there are `color(blue)(36)-color(magenta)3=color(brown)33` possible outcomes for which the sum is less than 11.

Since each outcome is (assumed to be) equally probable,
`color(white)("XXX")P("sum" < 11) = (color(brown)(33))/(color(blue)(36))=11/12`