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?

(I sent 2b as a picture.)

3. a) A checker is placed on a game board as shown below. Determine the numbers of paths the checker may take to get to each allowable square on the board if it can if it can move only diagonally forward one square at a time. *Note: The move starts for the red box. b) Expend: ( x+ y) + = c) If (x + y)10 , find the value of k at x* y2k. d) Find n and & at () + (3) - CO)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