Question

hans is playing a carnival game. he is blindfolded while he throws a dart at a board of 12 balloons, as shown below. each balloon is labeled \s\ for small, \m\ for medium, \l\ for large, or \xl\ for extra - large. the board has 1 small balloon, 8 medium balloons, 2 large balloons, and 1 extra - large balloon. he hits one of the balloons at random and wins a stuffed animal of that size.
(a) find the odds against hans winning a large stuffed animal.
(b) find the odds in favor of hans winning a large stuffed animal.

Explanation:

Step1: Calculate the number of favorable and unfavorable outcomes

• Total number of balloons \(n = 12\).
• Number of large balloons (favorable for winning a large stuffed animal) \(f=2\).
• Number of non - large balloons (unfavorable for winning a large stuffed animal) \(u = 12 - 2=10\).

Step2: Calculate odds against

• Odds against an event \(E\) is given by the ratio of the number of unfavorable outcomes to the number of favorable outcomes.
• Odds against winning a large stuffed animal \(=\frac{u}{f}=\frac{10}{2}=\frac{5}{1}\) or \(5:1\) or \(5\) to \(1\).

Step3: Calculate odds in favor

• Odds in favor of an event \(E\) is given by the ratio of the number of favorable outcomes to the number of unfavorable outcomes.
• Odds in favor of winning a large stuffed animal \(=\frac{f}{u}=\frac{2}{10}=\frac{1}{5}\) or \(1:5\) or \(1\) to \(5\).

Answer:

(a) \(5:1\) (or \(5\) to \(1\) or \(\frac{5}{1}\))
(b) \(1:5\) (or \(1\) to \(5\) or \(\frac{1}{5}\))