Question

Abdul 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 one small balloon, two medium balloons, two large balloons, and an extra large balloon. He hits one of the balloons random and wins a stuffed animal of that size. (a) find the odds in favor of Abdul winning and extra large stuffed animal. (b) find the odds against Abdul winning an extra large stuffed animal .

Answer 1

We have balloons that Abdul throw darts at.

The probabilities of winning a small, medium, large or extra large animal is equal to the quotient between the number of balloons of each type and the total number of balloons.

The odds of an event Y is equal to the probability of the event Y divided by the probability of the event Y not ocurring. It is usually represented as a ratio:

`\text{Odds}=\frac{P(Y)}{P(\text{not Y})}`

a) We have 7 XL balloons out of a total of 12 balloons.

So the probability of hitting an XL balloon is:

`P(XL)=(7)/(12)\approx0.583`

The probability of not winning an extra large stuffed animal are equal to the quotient between the number of balloons that are not XL and the total number of balloons.

As we have 7 XL balloons, we will have 12-7=5 not-XL balloons.

Then, the probability of hitting a non-XL balloon are:

`P(notXL)=(5)/(12)\approx0.417`

Then, the odds of winning a XL stuffed animal are:

`\text{odds in favor}=(P(XL))/(P(notXL))=((7)/(12))/((5)/(12))=(7)/(5)`

The odds agains him can be expressed as the inverse of the odds in favor:

`\text{odds against}=\frac{P(notXL)}{P(XL)_{}}=\frac{1}{\text{ odds in favor}}=((5)/(12))/((7)/(12))=(5)/(7)`

Answer:

a) Odds in favor: 7/5 or 7:5 (NOTE: we can express them either way)

b) Odds against: 5/7 or 5:7.