Geometric ProbabilityDate: 05/23/2001 at 12:38:29 From: Olivia Subject: geometric probability I am supposed to figure out the probability of an arrow hitting the bull's-eye of a target. I know that the target is a circle that is 3 feet in diameter and the bull's-eye is 6 inches in diameter. The problem also says that the arrow is equally likely to hit any spot on the target. How do I solve this? Date: 05/23/2001 at 12:41:55 From: Doctor Annie Subject: Re: geometric probability Hi, Olivia. Let's start with the basic idea of probability, which is used to calculate the "likelihood" that something will happen in a given situation. We find this by dividing the number of "favorable" outcomes, meaning what you want to happen, by the number of total outcomes, meaning all of the things that might happen. number of favorable outcomes probability = ------------------------------ number of possible outcomes For example, if we want to know the probability of rolling a 5 with a standard six-sided die, we know that there is only one favorable outcome (5), while there are 6 total outcomes (one for each side of the die). This means that the probability of rolling a 5 is 1/6. If we wanted to know the probability of rolling an odd number, we first count the favorable outcomes - 1, 3, or 5. There are 3. There are still 6 total possible outcomes, so the probability of rolling an odd number is 3/6, or 1/2. (If you don't know much about probability and want to learn more, check out the Dr. Math Introduction to Probability at http://mathforum.org/dr.math/faq/faq.prob.intro.html .) "Geometric probability", which is what your target problem is about, is exactly the same idea, except that we are dealing with the areas of regions instead of the "number" of outcomes. The equation becomes area of favorable region probability = ------------------------------ area of total region A typical problem might be this: If you are throwing a dart at the rectangular target below and are equally likely to hit any point on the target, what is the probability that you will hit the small square? 25 cm ------------------------------- | | | 5 cm | | ----- | | | | | 10 cm | | | | | ----- | | | ------------------------------- To solve this, we need to find the area of the favorable region, which is the small square, and the area of the total region, which is the rectangle. * The area of the square is (5 cm)^2, or 25 cm^2. * The area of the rectangle is 10 cm * 25 cm, or 250 cm^2. favorable 25 cm^2 * probability = ----------- = ---------- = 1/10. total 250 cm^2 This means that there is a 1 in 10 chance that a dart thrown at the rectangle will hit the small square. Try using these ideas to tackle your problem, and be sure to write back if you need more help! - Doctor Annie, The Math Forum http://mathforum.org/dr.math/ |
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