You have 8 pairs of mittens, each a different color. Left and right mittens are distinct. Suppose that you are fostering (possibly imaginary) kittens, and you leave them alone for a few hours with your mittens. When you return, you discover that they have hidden 4 mittens! Suppose that your kittens are equally likely to hide any 4 of your 16 distinct mittens. Let X be the number of complete, distinct pairs of mittens that you have left. (a) Compute the probability mass function of X, px(k). (Hint: You may want to first figure out the range Ωx).
(b) Find E[X] using your answer to part (a) and the definition of expectation.
(c) Find E[X] again, but using linearity of expectation this time and indicator RVs. Are these indicator RVs independent? Rhetorical (answering this next part is optional): Which method was easier? What if I increased the number of pairs of mittens?

b is the answer that makes most sense

step-by-step explanation:

14

step-by-step explanation:

14 x 3 = 42

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.

the answer is 11.2 (i know i’m late)