Question

4{ 73%. 12:46 PM hw2 - Read-only Read Only - You can't save changes to this file ECE 3U2 Homework z Due date: January 24, 2019 (Thursday), before clas:s 1. (10 points) A lot of 100 items contains k defective items. M (Ms100) items are chosen at random and tested (a) (2 pts) How many different ways can we choose M items from 100 items in the lot? (b) (4 pts) How many different ways that among the M items chosen, m (msk) are found defective? (c) (2 pts) Based on (a) and (b), what is the probability that m are found defective among the M items chosen? (d) (2 pts) A lot is accepted if no more than 1 of the Mitems are defective. What is the probability that the lot is accepted? 2" (10 points) Assume that 50% of the population are women and exactly 50% are men. (a) (5 pts) Jane (assuming a woman) has only one sibling. What is the probability that this sibling is female? (b) (5 pts) Jack (assuming a man) has only one younger sibling. What is the probability that this sibling is male? Hint: Both probabilities are conditional probabilities given on information known 3. (10 points) Consider a nonsymmetric binary communication channel with the following error probabilities: - If a 0 is sent, a 1 is detected at the receiver with a probability of 1 - If a 1 is sent, the receiver detects a 0 with a probability of £z Assume the input is 0 with probability p and 1 with probability 1-p (a) (2 pts) Find the probability that the output is 1. (6 pts) (b) (3 pts) Find the probability that the input was 0 given that the output is 1. (c) (3 pts) Find the probability that the input was 1 given that the output is 1 (d) (2 pts) Based on (b) and (c), if &0.02, £2 -0.05, p 0.5, which input is more probable given that the output is 1? (10 points) You have a fair five-sided die. The sides of the die are numbered from 1 to5 Each die roll is independent of all others, and all faces are equally likely to come out on top when the die is rolled. Suppose you roll the die twice. Consider the following events: 4. A: the total of two rolls is 10 B: at least one roll resulted in 5 C: at least one roll resulted in 1 (a) (6 pts) Calculate P[A], P[B], P[C] (b) (2 pts) Is event A independent of event B? (c) (2 pts) Is event A independent of event C? Please turn over

Answer 1

1(a)

Number of ways of choosing M items out of 100 is

/100 100!

(b)

There are k defective so number of ways of choosing m defective items out of k and M-n non defective items out of 100-k non defective items is

(100 - k)!

(c)

Let X shows the number of defective items out of M. The required probability is

k (100-k (100-k! 100 100! M 100-M)!

(d)

100- 100