You are given 4 to 1 odds against tossing three tails with three coins, meaning you win $4 if you succeed and you lose $1 if you fail. find the expected value (to you) of the game

-3/8

Step-by-step explanation:

The expected value will be equal to the probability of each of the results multiplied by its earning.

Here we have two possible results: win or lose. So, we need the probability of winning, that is, the probability that tossing three coins I get three tails, and the probability of losing, which will be the probability of getting anything different to three tails.

So, what is the probability of getting three tails?

We toss our first coin, we need to get a tail, The probability, assuming is a fair coin, will be 1/2.

Then we toss the second coin and we need that, having the first being tail, the second also need to be a tail. As the probabilities of both tossings are independent, this is, a tail in the 1st does not affect the probability of having a tail in the 2nd, the probability of having a tail in the second and the first is: (1/2) * (1/2).

Then, as the third coin also is independent to the first two, the probability of getting a tail in the third is: (1/2) * (1/2) * (1/2) = 1/8

So, if the probability of winning is 1/8 and the prize for winning is $4, the expected value of winning is:

EW = 1/8 * 4 = 4/8 = 1/2

Now, we need to do the same with the loses. The probability of losing is automatic, is 1 - probability of winning: 1 - 1/8 = 7/8. This is easy, if I do not win, I lose, and those are to only possible results, so both probabilities need to sum 1 (100%). And if I lose the game I lose $1, so the "prize" is -$1. Thus, the expected value of losing is:

EL = (7/8) * (-1) = -7/8

So, the expected value of the game is the sum of expected values of wining and losing:

ExpectedValue = 1/2 - 7/8 = 4/8 - 7/8 = -3/8

So, the expected value of playing such a game is - $3/8

the height of the surface increases from the center out to the sides of the road.

step-by-step explanation:

as we move out we move away from the x-axis.

and as mentioned in the problem statement that height of a road increases when we move away from the center x.

hence proved that 3rd option

the height of the surface increases from the center out to the sides of the road.

is the correct option.