Aparticle moves among n+1
vertices that are situated on a circle in the following manner. at each step it moves one step either in the clockwise direction with probability p or the counterclockwise direction with probability q=1−p. starting at a specified state, call it state 0, let t be the time of the first return to state 0. find the probability that all states have been visited by time t.
hint: condition on the initial transition and then use results from the gambler's ruin problem.

The value of x in this system of equations is 1. 3x + y = 9 y = –4x + 10 substitute the value of y in the first equation: combine like terms: apply the subtraction property of equality: apply the division property of equality: 3x + (–4x + 10) = 9 –x + 10 = 9 –x = –1 x = 1 what is the value of y?

Which term describes the point where the perpendicular bisectors of the three sides of a triangle intersect?

Aprism and two nets are shown below: image of a right triangular prism and 2 nets. the triangle bases have base 4, height 3, and diagonal side 5. the length of the prism from the bases is 9.4. net a has 3 rectangles in a row. the middle one has sides ac and cd. this middle one has triangle abc on top and identical triangle below. net b is the same except that the triangle base abc is on top of the last rectangle which has sides ac and cd. units are in inches. part a: which is the correct net for the prism? explain your answer. (2 points) part b: write the measurements of sides ab, bc, and cd of the correct net. (4 points) part c: what is the surface area of the prism? show your work. (4 points)