1. Suppose we throw a dart at a square target 1 meter on a side, which has a bullseye in the center of radius 0.1 meters. Assume the dart lands with equal probability anywhere inside the square target. Give the probabilities of the following events:

(a) The dart lands in the bullseye;

(b) The dart lands within 0.1 meter of an edge of the square target (i.e., the shortest distance from the dart to the closest edge is 0.1);

(c) The dart lands within 0.1 meter of a corner of the square target (i.e., the distance from the dart to the closest corner point is 0.1);

(d) The dart lands in the exact center of the square (equivalently, in the exact center of the bullseye).

2. Consider the random experiment of flipping a fair coin until a head appears. The result of the random experiment is the number of flips. Let A = "it takes an odd number of flips." Give the probability P(A).Show reasoning,(Hint: compare the sequence of probabilities in the case of an odd number of flips, and the sequence of probabilities in the case of an even number.)

Question 3:Suppose that each time Lamar charges an item to his credit card, he rounds the amount to the nearest dollar in his records (assume that for x dollars, the amount x.50 is rounded to x + 1 dollars). The round-off error is defined as (recorded - actual); the units are dollars, so if Wayne charges $4.25, he records it as $4, and the round-off error is $-0.25, but if he charges $4.75, the value recorded is $5 and the round-off error is $0.25. Assume this is random, so that each time Lamar charges to his card, he performs a equiprobable random experiment whose outcome is the round-off error. Let event A = "at most 3 cents is rounded off in either direction" (i.e., | recorded - actual | 0.03). Analyze.

4. Consider the formula: P( ( A Bc ) ( Ac B ) ) = P(A) + P(B) - 2*P(A B). (This is the "symmetric difference" that I mentioned at the very first lecture.)

(a) Draw a Venn Diagram illustrating this formula and explain in words why it is true.