Problem 30 - Algebra for College Students

Question

If the probability that a Sidewinder missile hits its target is \(\frac{8}{9},\) then what are the odds a) in favor of the Sidewinder hitting its target? b) against the Sidewinder hitting its target?

Step-by-Step Solution

Answer

a) Odds in favor are 8:1. b) Odds against are 1:8.

Step 1: Understanding Probability

The probability of an event is the ratio of favorable outcomes to the total number of possible outcomes. Here, the probability that the Sidewinder hits its target is given as \(\frac{8}{9}\).

Step 2: Calculate Probability of Not Hitting

To compute the probability of the Sidewinder not hitting its target, subtract the hitting probability from 1. \ P(\text{not hitting}) = 1 - P(\text{hitting}) = 1 - \(\frac{8}{9}\) = \(\frac{1}{9}\).

Step 3: Determine Odds in Favor

Odds in favor are given by the ratio of the probability of the event occurring to the probability of the event not occurring. \ \text{Odds in favor} = \(\frac{P(\text{hitting})}{P(\text{not hitting})}\) = \(\frac{8/9}{1/9}\) = 8:1.

Step 4: Determine Odds Against

Odds against are given by the ratio of the probability of the event not occurring to the probability of the event occurring. \ \text{Odds against} = \(\frac{P(\text{not hitting})}{P(\text{hitting})}\) = \(\frac{1/9}{8/9}\) = 1:8.

Key Concepts

Probability

Probability is a way to measure the likelihood of an event happening. You can think of it as a fraction. The numerator is the number of favorable outcomes, and the denominator is the total number of possible outcomes. For example, if the probability that a Sidewinder missile hits its target is \(\frac{8}{9}\), that means out of 9 attempts, it is expected to hit 8 times. This fraction tells us how likely it is to happen.
In our exercise, we are given this probability to determine the odds in favor and against the missile hitting its target.

Odds in Favor

Odds in favor refer to the ratio of the probability of an event occurring to the probability of it not occurring. In our Sidewinder missile problem, we know the probability of hitting is \(\frac{8}{9}\).
To find the odds in favor, we need both the probability of hitting and the probability of not hitting. The probability of not hitting is calculated as \(\frac{1}{9}\).
Now, we can compute the odds in favor. It’s the ratio of these two probabilities: \[\text{Odds in favor} = \frac{P(\text{hitting})}{P(\text{not hitting})} = \frac{8/9}{1/9} = 8:1.\]
So, for every 1 time it does not hit, the Sidewinder is expected to hit 8 times. This tells us the event hitting is very likely compared to not hitting.

Odds Against

Odds against are essentially the flip side of odds in favor. They measure the ratio of the event not occurring to the event occurring. In our Sidewinder missile example, we already calculated the probability of it not hitting as \(\frac{1}{9}\).
By reversing these probabilities, we get the odds against. We simply use this formula: \[\text{Odds against} = \frac{P(\text{not hitting})}{P(\text{hitting})} = \frac{1/9}{8/9} = 1:8.\]
This means for every 8 times the missile hits, it is expected to miss 1 time. Knowing the odds against helps us understand how unlikely the event of not hitting is.

Event Occurrence

An event occurrence refers to the actual happening of an event. It can be described using its probability.
In our problem, the occurrence is the Sidewinder missile hitting its target. We often start by calculating the probability of the occurrence and then use it to find related measures like odds in favor and against.
When the probability of an event is given or calculated, it sets a foundation for further statistical evaluations. Whether an event did happen (like hitting the target) or did not (missing the target), probability and odds give us a quantifiable way to understand and predict future behaviors.