   SEARCH HOME Math Central Quandaries & Queries  Subject: SAT question Name: rose Who are you: Parent The radius of the a small circle is 1 and the radius of the large circle is 2. If two points are selected at random from the interior of the large circle, what is the probability that both points will be from the shaded region? the shaded area outside the small circle. Hi Rose,

The area of a circle is π r2 where r is the radius, hence the area of the large circle is 4 π and the area of the small circle is π. Thus the area of the shaded region is 4 π - π = 3 π. Hence the area of the small circles is 1/4 of the area of the large circle and the area of the shaded region is 3/4 of the area of the large circle.

If you chose a point at random from the large circle the probability that it is in the shaded region is 3/4 and then if you independently again chose a point from the large circle at random the probability that it is in the shaded region is 3/4. Hence the probability that both points are in the shaded region is 3/4 3/4 = 9/16.

Penny

Rose,

Isn't this an example of an undefined problem? The random process can be linear (choose a random angle, then a random radius from 0 to 2 -- there's a 50-50 chance the chosen point will land outside the inner circle), or quadratic (choose a random piece of the area -- like shooting at a target in which case the probability that the bullet lands outside the inner circle is 3/4).

Chris

Hi again Rose,

Chris is correct, the random process is not described and hence the problem can't be answered in a unique way. I assumed the "shooting at a target" process to arrive at my answer of 9/16 for the probability. I expect that this is what the SAT exam composers wanted for an answer but the problem is not well defined and there is more than one correct answer depending on the random process used.

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.