a The center of the coin has to be within the coin's radius of the outer edge of the circle. Use the inner circle that is formed to set up the probability.
B
b Use the answer from Part A as a percent to determine how many coins out of 100 would land in the desired zone.
A
a 1.4 %
B
b 100 Explanation: See solution.
Practice makes perfect
a To find the probability that the coin will land entirely in the circle, we will need the area of the circle and the area of the region. Let's look at that first.
The whole target is the square with an area of 8* 8= 64 in^2. The favorable area, is not simple the area of the circle. The coin has to lie entirely with in the circle. Therefore, its center needs to land within a smaller inner circle.
Since the coin's radius is 1532, the center of our coin has to be more than that distance from the circles edge. That means the center must lie within the circle of radius 1- 1532= 1732. Let's find the area of that circle.
We can now find the probability that the coin will land completely within that circle by look at the ratio of the inner circle's area to area of the square.
A_(Inner Circle)/A_(Square) = 0.8866/64
Therefore, the probability that a quarter lands entirely in the circle in one random toss is about 1.4 %.
b From Part A, we learned that the probability of a quarter landing entirely in the circle is about 1 %.
This percentage means that if the coins randomly fall within the square, 1 in every 100 would land entirely within the circle. Therefore, we must toss 100 coins to expect a prize.
