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Homework 1 1. Let A = {1, 2, 3, 4, a, b, c}, B = {1, 2, 5, 6, b, c, d, e, f }, C = {1, 2, a, d, g, h}. (a) Is 1 A? Is {1} A? Is {1, 2} A? (b) Compute...

I wonder from Q4 to Q6 an Q3 (d).
Homework 1 1. Let A = { 1 , 2 , 3 , 4 ,a,b,c } , B = { 1 , 2 , 5 , 6 ,b,c,d,e,f } , C = { 1 , 2 ,a,d,g,h } . (a) Is 1 A ? Is { 1 } ∈ A ? Is { 1 , 2 } ⊆ A ? (b) Compute A B and A B . (c) Compute ( A B ) C and A ( B C ). Are they equal? (d) Show ( A B ) C = ( A C ) ( B C ) and A ( B C ) = ( A B ) ( A C ) for these particular choices of A , B and C . (e) Define A - B = A B 0 . Compute A - B and B - A . Is A B = ( A - B ) ( B - A ) ( A B )? (f) What are # A , # B , #( A B ). How to use these three quantities to compute #( A B )? 2. How many possible outcomes are there if one flips a fair coin 5 times. How many of them have two heads and three tails. Let X be the number of head in an outcome. Based on what you get, compute the probability of X equal to 2 say P [ X = 2]. Furthermore, compute P [ X = 4]. 3. Flip a fair dice 3 times. (a) What is the probability of “no 6 observed”. (b) What is the probability of “at least one 6 observed”. (c) What is the probability of “exactly one 6 observed”. (d) What is the probability of “at most two 6 observed”. (e) What is the probability of “at least two 6 observed”. 4. A bag has 10 blue balls and 8 red balls. Randomly choose 3 of them. (a) Compute the probability of the event that there is exactly one red ball, if the choice is without replacement and if the choice is with replacement. (b) Compute the probability of the event that there is at most one red ball, if the choice is without replacement and if the choice is with replacement. 5. Flip a dice 4 times. Let X be the total number of 5 or 6. (a) Compute the probability of X = 0, i.e., there is neither 5 nor 6. (b) Compute the probability of X = 1, i.e., there is exactly one of 5 or 6? (c) Compute the probability of X = 2. (d) Compute the probability of X < 2 . 5. (e) Compute the probability of X > 2. 6. There are 10 football tickets with seat number from 1 to 10 in a row. Tom and Jerry are friends and they do not want to be apart. If they are asked to randomly pick up their tickets, what is the probability for Tom and Jerry to sit next to each other. 1
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