1.involving the flipping of a loaded coin, but assume that Pr[H]=0.7 Also, assume that the coin is flipped 4 times, and the random variable Xis defined to be 4 times the number of heads minus 3 times the numberof tails.How many different values are possible for the random variable? Fill in the table below to complete the probability density function. Be certain to list the values ofX in ascending order. Value of X Probability 2. involving a defective vending machine. Assume that the machine yields the item selected 85 percent of the time, and returns nothing 15 percent of the time. Three individuals attempt to use the machine. LetXbe defined as the number of individuals who obtain the item selected. How many different values are possible for the random variable X? Fill in the table below to complete the probability density function. Be certain to list the values ofxin ascending order. Value of X 0 1 2 3 Probability 3. involving selecting balls from a box of numbered balls. Assume that the box contains balls numbered from 1 through 8, and that 3 are selected. A random variableXis defined as 2 times the number of odd balls selected, plus 3 times the number of even. How many different values are possible for the random variable ? Fill in the table below to complete the probability density function. Be certain to list the values of Xin ascending order. Value of X Probability 4. involving the flipping of a weighted coin. Assume that the coin is weighted so that Pr[H]=3/11and that the coin is flipped until a head appears, or 3 consecutive tails appear. The random variable Xis defined to be the total number of flips of the coin. How many different values are possible for the random variable ? Value of X Probability 5. Assume that the test has 4 true/false questions, and 12 multiple choice questions (5 options per questions), each with exactly one correct answer. If a student guesses randomly on each question, what is her expected score? 6. involving the expected number of hits for a baseball player during a certain month. Assume that the batter hits with probability 0.3 against left-handed pitchers, and with probability 0.15 against right-handed pitchers. Both situations are Bernoulli processes. One month the player bats 17 times against lefthanded pitchers and 76 times against right-handed pitchers.How many hits should the batter expect to get during this month 7. in which three balls are selected simultaneously and randomly from a box. But assume that the box contains 4 red balls and 5 green balls. Also assume that each red ball selected pays $0.30, that each green ball selected pays $0.40, and that it costs $1.40 to play the game. A random variable Xis defined to be the net payoff for a player. How many different values are possible for the random variable? Fill in the table below to complete the probability density function. Be certain to list the values ofXin ascending order. Value of X Probability 8. involving the selection of two numbers. Assume that you select two 3-digit numbers at random from the set of consecutive integers from 000 through 999. The selections are made with replacement and are independent of one another. If the two numbers are the same, you make $600. If they are different, you lose $5. A random variableX is defined as your gain or loss.How many different values are possible for the random variableX Fill in the table below to complete the probability density function. Be certain to list the values ofXin ascending order. Enter a loss as a negative number. Value of X 1 2 Probability 9. Assume that the pond contains 110 fish: 82 red, 27 orange, and 1 purple. A contestant pays $0.55 to randomly catch one fish and receives $0.35 if the fish is red, $1.05 if the fish is orange, and $14.00 if the fish is purple. How much (on average) does the carnival gain on each play