1. Suppose two balanced coins are tossed and the upper faces are observed.

a. List the sample points for this experiment.

b. Assign a reasonable probability to each sample point. (Are the sample points equally likely?)

c. Let A denote the event that exactly one head is observed and B the event that at least one head is observed. List the sample points in both A and B.

d. From your answer to part (c), find P(A), P(B), P(A ∩ B), P(A ∪ B), and P(A ∪c)

2. Because market interest rates were near all-time lows at 4% per year, a hand tool company decided to call (i.e., pay off ) the high-interest bonds that it issued 3 years ago. If the interest rate on the bonds was 9% per year, how much does the company have to pay the bond holders? The face value (principal) of the bonds is $6,000,000.

3. It is known that a patient with a disease will respond to treatment with probability equal to 0.9. If three patients with the disease are treated and respond independently, find the probability that at least one will respond.

4. Define the following events:

A: At least one of the three patients will respond.

B1: The first patient will not respond.

B2: The second patient will not respond.

B3: The third patient will not respond.

Then observe that ¯ = B1∩B2∩B3

P(A)=1-P(A’)= 1-P(B1∩B2∩B3) and ¯ Applying the multiplicative law, we have where, because the events are independent,

Substituting we obtain P(A) = 1 − (.1)3 = .999. Notice that we have demonstrated the utility of complementary events. This result is important because frequently it is easier to find the probability of the complement, P(A), than to find P(A) directly.

5. A continuous-time speech signal xa(t) is sampled at a rate of 8 kHz and the samples are subsequently grouped in blocks, each of size N. The DFT of each block is to be computed in real time using the radix-2 decimation-in-frequency FFT algorithm. If the processor performs all operations sequentially, and takes 20 μs for computing each complex multiplication (including multiplications by 1 and −1) and the time required for addition/subtraction is negligible, then the maximum value of N is _

6. A voice-grade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of 4.0 kHz and two-sided noise power spectral density η 2 = 2.5 × 105 Watt per Hz. If information at the rate of 52 kbps is to be transmitted over this channel with arbitrarily small it error rate, then the minimum bit-energy Eb (in mJ/bit) necessary is