Sure, let's go through each part of the problem step by step.
### (a) Probability of Deandre winning a prize
Deandre's odds in favor of winning a prize are given as [tex]\( \frac{4}{3} \)[/tex].
Step 1: Understanding odds in favor
The odds in favor of an event [tex]\( A \)[/tex] happening is given by the ratio [tex]\(\frac{p}{q}\)[/tex], where [tex]\( p \)[/tex] represents the number of favorable outcomes, and [tex]\( q \)[/tex] represents the number of unfavorable outcomes. In this problem, the odds in favor are [tex]\( 4:3 \)[/tex].
Step 2: Convert odds to probability
The probability of the event occurring is given by:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \][/tex]
So, we need to sum the favorable and unfavorable outcomes:
[tex]\[ \text{Total outcomes} = 4 + 3 = 7 \][/tex]
Then, the probability [tex]\( P \)[/tex] is:
[tex]\[ P = \frac{4}{7} \][/tex]
So, the probability of Deandre winning a prize is:
[tex]\[ \boxed{0.5714285714285715} \][/tex]
### (b) Odds against a new customer ordering vanilla ice cream
The probability of a new customer ordering vanilla ice cream is [tex]\( \frac{9}{17} \)[/tex].
Step 1: Understanding the probability provided
The probability [tex]\( P \)[/tex] of an event (ordering vanilla ice cream) is given by:
[tex]\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \][/tex]
In this problem, [tex]\( P \)[/tex] is given as [tex]\( \frac{9}{17} \)[/tex].
Step 2: Calculate the odds against the event
The odds against an event [tex]\( A \)[/tex] occurring are given by:
[tex]\[ \text{Odds against} = \frac{\text{Number of unfavorable outcomes}}{\text{Number of favorable outcomes}} \][/tex]
The number of unfavorable outcomes is [tex]\( 1 - P \)[/tex]:
[tex]\[ 1 - P = 1 - \frac{9}{17} = \frac{17}{17} - \frac{9}{17} = \frac{8}{17} \][/tex]
So the odds against a new customer ordering vanilla ice cream are:
[tex]\[ \text{Odds against} = \frac{\frac{8}{17}}{\frac{9}{17}} = \frac{8}{9} \][/tex]
Thus, the odds against a new customer ordering vanilla ice cream are:
[tex]\[ \boxed{0.8888888888888888} \][/tex]
These are the step-by-step solutions to the given problems.
