For all the questions, please show the work (steps), also show diagrams, pictures, etc. And formulas)

1a) Suppose you were to toss three coins. What is the likelihood that you would observe at least two heads? What is the expected number of heads?

*Note: In the solution must be shown: the tree diagram, the table of the theoretical probability distribution, the histogram

2a) Suppose you want to select three pencils from a set of four Red and three Blue pencils.

i) What is the probability that at least one Red pencil is selected?

ii) What is the expected numberof Red pencils in the selection?

2b.

b) Which of the following are valid probability distribution? Explain the answer. P(X) X P(X) P(X) 0.15 0.5 0.45 0.4 0.25 0.3 0.38 0.25 0.25 0.2 0.17 0.25 0.2