Alice is trying to transmit to bob the answer to a yes-no question, using a noisy channel. she encodes "yes" as 1 and "no" as 0, and sends the appropriate value. however, the channel adds noise; specifically, bob receives what alice sends plus a n(0,σ2) noise term (the noise is independent of what alice sends). if bob receives a value greater than 1/2 he interprets it as "yes"; otherwise, he interprets it as "no". (a) find the probability that bob understands alice correctly. (b) what happens to the result from (a) if σ is very small? what about if σ is very large? explain intuitively why the results in these extreme cases make sense.

Ben wants to play a carnival game that costs $2. in his pocket he has 5 red tickets worth 35 cents each, and 15 blue tickets worth 10 cents each. which of the following systems of inequalities correctly represents the constraints on the variables in this problem? let represent the red tickets and b represent the blue tickets.

Which statement accurately explains whether a reflection over the y axis and a 270° counterclockwise rotation would map figure acb onto itself?

Submarines control how much they float or sink in the ocean by changing the volume of air and water contained in large ballast tanks. when the tanks are completely full of water, the submarine increases its overall mass and sinks down to the bottom. when the tanks are completely full of air, the submarine reduces its overall mass and floats to the surface. depending on the density of the seawater surrounding the submarine, it will pump seawater in or out of the tanks in order to achieve the same overall density as the sea water and float neutrally in the water. the volume of the submarine never changes. when the tanks are completely full of water, a submarine with a volume of 7.8\times10^3\text{ m}^37.8×10 3 m 3 7, point, 8, times, 10, start superscript, 3, end superscript, space, m, start superscript, 3, end superscript has a total mass of 8\times10^6\text{ kg}8×10 6 kg8, times, 10, start superscript, 6, end superscript, space, k, g. the density of the seawater is 10^3\text{ kg/m}^310 3 kg/m 3 10, start superscript, 3, end superscript, space, k, g, slash, m, start superscript, 3, end superscript. to make that submarine float neutrally, and neither float nor sink in the ocean, what volume of water does that submarine need to subtract from its tanks?