Consider the system of components connected as in the accompanying picture. Compo- nents 1 and 2 are connected in parallel, so that subsystem works iff either 1 or 2 works; since 3 and 4 are connected in series, that subsystem works iff both 3 and 4 work. Sup- pose the components work independently of one another with probabilities p1, p2, p3, and p4:

The yellow and green dice are tossed together. If A is the event that a 2 is shown on the yellow die and B is the event that 3 is shown on the green die, thena) determine P(A), P(B)b) Find the probability of getting a 2 on the yellow die and a 4 on the green.

Each of the numbers 0 - 9 is written on a sheet of paper and the ten sheets of paper are placed in a box If two of these sheets of paper are selected from the box at random, without replacement, find the probability that both numbers are greater than 7.

Write the m.g.f. of a r.v. X taking on values -4, 0 and 4 with respective probabilities 0.3, 0.4, and 0.3. 0.3 e4 +0.4 • e² + 0.3 -4z 0.3 e +0.4+ 0.3. e4z 1+0.3. e4z +0.3 · e-4z e4 0.3 ez +0.4 e² + 0.3. e4z

A favorite casino game of dice “craps” is played in the following manner: A player starts by rolling a pair of balanced dice. If the roll (the sum of two numbers showing on the dice) results in a 7 or 11, the player wins. If the roll results in a 2 or 3 (called “craps”) the player loses. For any other roll outcome, the player continues to throw the dice until the original roll outcome recurs (in which case the player wins) or until a 7 occurs (in which case the player loses). When answering the following questions, you can use this outcome chart for the roll of two dice: Provide the probability answers in fraction and in decimal forms rounded to 4 digits.     A. List the possible outcomes (sample space) for winning on the first roll of the dice.   B. What is the probability that a player wins the game on the first roll of the dice?   C. List the possible outcomes (sample space) for losing on the first roll of the dice.   D. What is the probability that a player loses the game on the…

Two unbalanced dice are thrown at the same time. The first die has the property that the probability of getting a side of 1 and 6 is three times the other side. While the second dice has the property that the probability of the appearance of the sides of the eye 3 and 4 is three times the probability of the other side. If X represents the appearance of the side of the first die and Y represents the appearance of the side of the second die. a. Find the joint probability mass function, P(x,y) b. Find the X and Y marginal probability mass functions c. Find E(X), E(Y), and E(XY)