Step 1. The wheel has 10 slices:
7 slices are black
2 slices are grey
1 slice is white
Required: Find the odds against and the odds in favor of Kareem winning a gift card if he wins it if the arrow stops on a black slice.
Step 2. The odds are defined as follows:
[tex]\begin{gathered} Odds\text{ against }=\frac{Number\text{ of unfavorable outcomes}}{Number\text{ of favorable outcomes}} \\ Odds\text{ in favor}=\frac{Number\text{ of favorable outcomes}}{Number\text{ of unfavorable outcomes}} \end{gathered}[/tex]Step 3. Solving part (a)
Remember that he wins it if the arrow stops on a black slice, therefore, the number of unfavorable outcomes are all of the slices that are not black which are 3:
[tex]Number\text{ of unfavorable outcomes: 3}[/tex]And the number of favorable outcomes is the number of black slices:
[tex]Number\text{ of favorable outcomes: 7}[/tex]the odds against are:
[tex]\begin{gathered} Odds\text{ aga}\imaginaryI\text{nst}=\frac{Number\text{ of unfavorable outcomes}}{Number\text{ of favorable outcomes}} \\ \downarrow \\ Odds\text{ aga}\imaginaryI\text{nst}=\frac{3}{7} \end{gathered}[/tex]Step 4. Using the same information, we calculate the odds in favor to solve part (b):
[tex]\begin{gathered} Odds\text{ }\imaginaryI\text{n favor}=\frac{Number\text{ of favorable outcomes}}{Number\text{ of unfavorable outcomes}} \\ \downarrow \\ Odds\text{ }\imaginaryI\text{n favor}=\frac{7}{3} \end{gathered}[/tex]Answer:
[tex]\begin{gathered} (a)\text{ }\frac{3}{7} \\ (b)\text{ }\frac{7}{3} \end{gathered}[/tex]