Probability played a role in the rigging of the april 24, 1980, pennsylvania state lottery (los angeles times, september 8, 1980). to determine each digit of the three-digit winning number, each of the numbers 0, 1, 2, . . , 9 is placed on a ping-pong ball, the ten balls are blown into a compartment, and the number selected for the digit is the one on the ball that floats to the top of the machine. to alter the odds, the conspirators injected a liquid into all balls used in the game except those numbered 4 and 6, making it almost certain that the lighter balls would be selected and determine the digits in the winning number. then they bought lottery tickets bearing the potential winning numbers. how many potential winning numbers were there (666 was the eventual winner)?

There were 8 potential winning numbers

Step-by-step explanation:

According on what the exercise says, there are 2 numbers likely to be selected in each of the 3 rolls: 4 and 6. Since we are picking 3 digits, and each digit give us 2 possibilities, then we have a total of 2³ = 8 potential winners (namely 444, 446, 464, 644, 466, 646, 664 and the winner 666).

there is no grid

step-by-step explanation:

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.