A game is played where a biased coin is flipped. A head is twice as likely as a tail. It costs $5 to plat the game and if a head occurs you win &7 but if a tail occurs you pay $3 which of the following is correct?

The game is fair
The expected valve of net gain is positive and the player should play.

The expected valve of net gain is negative and the player should play

The expected valve of net gain is positive and the player should not play

The expected valve of net gain is negative and the player should not play

prob of tail --- x

prob of head --- 2x

x+2x = 1
3x=1
x = 1/3

expected win = (1/3)7 + (2/3)(-3) = 1/3 or $.33
but you are paying $5.00 for a return of .33, silly!

"The expected valve of net gain is positive and the player should not play "

A student is interested in earning some money to pay her tuition. A game is proposed that is very enticing. Roll two dice; pay $25 to play if a sum of 10 or larger occurs she pays $8; if a sum of 3 or smaller occurs she pays the amount of sum in dollars; if another sum occurs she earns the value of the sum in dollars

Which statement is correct ?

The game is fair

The expected valve of net gain is positive and player should play

The expected valve of net gain is negative and the player should play

The expected valve of net gain is positive and the player should not play

The expected valve of net gain is negative and the player should not play

Price to play the game = $25

Sum/return
2/2
3/3
4/4
5/5
6/6
7/7
8/8
9/9
10/8
11/8
12/8
So the maximum return is $9 (if she throws a sum of 9), can never recover the $25 she pays to play.
If she is the player, the recommendation is:
"The expected valve of net gain is negative and the player should not play"

She might make some money for college if she organizes the game.

To determine which of the options is correct, we need to calculate the expected value of net gain for playing this game.

First, let's calculate the probabilities of getting a head (H) and a tail (T). We are given that the probability of getting a head is twice that of getting a tail. Let's assign P(H) as the probability of getting a head and P(T) as the probability of getting a tail.

Since the sum of probabilities must be equal to 1, we have:
P(H) + P(T) = 1

Given that the probability of getting a head is twice that of getting a tail, we have:
P(H) = 2P(T)

Rearranging the equations, we get:
2P(T) + P(T) = 1
3P(T) = 1
P(T) = 1/3

Using this value, we can find P(H) as:
P(H) = 2P(T)
P(H) = 2(1/3)
P(H) = 2/3

Now let's calculate the net gain for each outcome:
If you get a head, you win $7. So the net gain would be $7 - $5 (the cost to play) = $2.
If you get a tail, you lose $3. So the net gain would be -$3 - $5 = -$8.

Let's calculate the expected value of net gain using the probabilities and net gains for each outcome:
Expected Value = P(H) * Net Gain (H) + P(T) * Net Gain (T)
Expected Value = (2/3) * $2 + (1/3) * (-$8)

Calculating this equation, we find:
Expected Value = $4/3 - $8/3
Expected Value = -$4/3

The expected value of net gain is negative, specifically -$4/3. Based on this calculation, the correct answer is:

"The expected value of net gain is negative and the player should not play."