Find the odds for and the odds against the event rolling a fair die and getting a 4 comma a 5 comma or a 2.
a. the odds for the event are
nothing to
nothing.

The odds in favor are 3:3 which reduces to 1:1

There are 3 ways to get what we want, which is 4, 5 or 2. There are 3 ways to get what we don't want (1,3,or 6). So we list those two values as a ratio like so 3:3, which reduces to 1:1 after you divide both parts by 3.

Here's another example. If we wanted to know the odds in favor of rolling a 5 or 6, then the answer would be 2:4 which reduces to 1:2. This is because there are 2 ways to get what we want (rolling a 5 or 6) and 4 ways to get what we don't want (rolling either 1,2,3, or 4). The number of events we favor is listed first when it comes to "odds in favor" problems.

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is