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1. A couple decides to have children until they have a boy. Let's assume that they are able to continue to do so and each time they conceive, they only have a single child (no twins or triplets, etc.). a. What is the probability that the first time that they have a boy is on their 4th child (that is, that they have 3 girls before they have a son)? b. If their first 3 children are girls, what is the probability that their next child is boy? c. On average, how many children should they expect to have to have before having a boy? (Hint: use math expectation) 2. We have a die that is numbered 3 with two 1's, two 2's, and two 3's. Let's assume it is fair (each side is equally probable) and that the die is rolled 3 times. And, let's assume that the 3 rolls are independent. Let X, Y, and Z be the outcomes of the first, second and third rolls, respectively. a. What is the probability distribution of X+Y+Z? That is, create a table that contains each unique possible value of X+Y (each value only listed once) and each possibility's corresponding probability. b. What is the probability that X+Y+Z is greater than or equal to 7? 3. We have a fair six-sided die numbered 2, 4, 6, 8 10, and 12. a. Find the math expectation of a single roll. b. Find the math expectation of the numerical sum of 5 rolls. c. Find the math expectation of the numerical product (i.e., multiplication) of 4 rolls. d. Find the cumulative distribution function of the outcome of a single die roll and draw its graph. 4. X1. X2, ... X144 are independent and identically distributed random variables such that E(x) = 12 and var(x) = 36. What is the standard deviation of their average? In other words, what is the standard deviation of X = X1+X2+ +X1442 144 5. If the cumulative distribution function of X is given by the function below, then find P (X <1.5). F(x) 0, if S o F(x) = =x if o x<2 F(x)=1,if x>2 6. A local pizza place has a profit of $1, $3, and $2 for each sale of their small, medium and large pizzas, respectively. In general, their small (personal size) and large (family size) are their most popular pizza sizes (with about 40% of their sales contributing to each of these sizes). Size Small Medium Large x $1 $3 $2 P 0.40 0.20 0.40 a. Find the math expectation of a single pizza sale. b. In a typical day, there are about 200 pizza sales. Find the math expectation of the profit for a single day (200 sales). c. Find the variance of the profit for a single sale. d. Find the standard deviation for a single sale. c. What might the pizza place do to increase their profit?

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