Find each of the following probabilities when n independent Bernoulli trials are carried out with probability of success p.

(a) the probability of no successes

(b) the probability of at least one success

(c) the probability of at most one success

(d) the probability of at least two successes

(e) the probability of no failures

(f) the probability of at least one failure

(g) the probability of at most one failure

(h) the probability of at least two failures

4. Complete the table below by solving each equation algebraically (expressing both sides with the same base) and graphically (using technology). a. 2041 =16 b. 93 = 27 C. 57- =254 Algebraically (2 marks) Algebraically (2 marks) Algebraically (2 marks) Graphically (1 mark) Graphically (1 mark) Graphically (1 mark) M = Coordinates of intersection Coordinates of intersection Coordinates of intersection Solution Solution Solution x =11:06 Lab manuals Putting it All Together Following the same format as the examples, write text descriptions for the algebra needed to arrive at each of the simplified derivatives below. It will likely help to work out the problems on a piece of paper. Algebra: Algebra: (x3. ex . (13 + 3x - 1) STEP 1 (the product of the first two factora): Algebra: STEP 2 ( the product of the third factor with the rest of the function): Algebra: (x - 2x+ 1) dx x - 1 Algebra: dx \\c + do Algebra:Definition 4.4 (Algebraic Closure). An algebraic closure of a field K is a field extension K C F such that F is algebraic over K and F is algebraically closed. M311 F20 FINAL PROJECT 5 Observation 4.5. We have R C C is an algebraic closure of R. We have C is algebraically closed. the field C is R(i) and i is a root of r' + 1 so C is algebraic over R. Definition 4.6 (Relative Algebraic Closure). Given a field extension K C F we have the set {o ( F : a is algebraic over K ). Corollary 6.2.8 of BB states that this set is a subfield of K. We denote this set by Kp. It is called the relative algebraic closure of K in F. (4) Problem Show that if K C F is a field extension and F is algebraically closed then Ky is an algebraic closure of K. We know it Kpis a field. By definition it is algebraic over K. The problem is to show it is algebraically closed. Definition 4.7. The field Q = Qc is an algebraic closure of Q. The field Qc is called the field of algebraic numbers.5. A number if algebraic if it is the root of a non-constant polynomial with integer coefficients. i. Show that 1 is algebraic. ii. Show that \\ 2 is algebraic. iii. Show that the set of algebraic numbers is countable. Deduce that there are real numvers that are not algebraic