I want to see how to solve the P1.1, P1.7, and P1.12 in the attachment.

Pl.l Formally prove the union bound which states that for any events A., A2. ..., AM (not necessarily mutually exclusive) M Pr 52mm]. i=1 M U A: i=1 Hint: Prove using mathematical induction. First show that the given result is true for M = 1 and M = 2. Then assume that it is true for M 1. Now. using the logical arguments used to show the validity for M = 2, show that the result is also valid for M. PL? Four fair dice. colored red. green. blue, and white, are tossed. (a) Determine the probability of getting all four face values equal to 3. (b) After tossing, a quick glance at the outcome indicated that two of the face values were 3 but no other information (about their color or the values of the remaining two faces] was noted. Now determine the probability of getting all four face values equal to 3. (c) After tossing it was further noted that, out of the two observed face values of 3, one was red in color. Now determine the probability of getting all four face values equal to 3. (d) After tossing, it was nally conrmed that the two observed face values of 3 were red and white. Now determine the probability of getting all four face values equal to 3. (9) You should get distinct answers for the above four probabilities. Qualitatively explain why the above four probabilities make sense. Anmrs: (b) lll7l, (c) U91. and (d) 1136. Pl.12 Random variable X is described by the cumulative distribution function (CDF) qu) shown below. inX) (a) Determine the probability distribution function (pdfl fx (1:) of X. Sketch fx (x) on the right above and indicate critical points to get full credit. (b) Determine the probability Pr [0 s X