In a popular casino​ game, you can bet one whether a ball will fall in an arc on a wheel colored​ red, black, or green. Say the probability of a red outcome is StartFraction 18/38 EndFraction 18/38​, that of a black outcome is StartFraction 18/ 38 EndFraction 18/38​, and that of a green outcome is StartFraction 2/38 EndFraction 2/38. Suppose someone makes a ​$20 bet on black. Find the expected net winnings for this single bet.

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Answer:

The expected net winnings for the bet are -$1.0526

Step-by-step explanation:

P(x =+$20) = P(Black outcome) = 18/38

P(x =-$20) = P(red outcome) + P(green outcome)

= 18/38 + 2/38 = 20/38

Hence the probability distribution of x = $20 , P(x) = 18/38

x = -$20, P(x) = 20/38

Expected value of the random variable x is given by ;

miu = Summation [xP(x)] = 20(18/38) - 20( 20/38)

= -$1.0526

hence, the expected net winnings for the bet are -$1.0526

This implies that if a player bet on a very large number of games, the player would on the average lose $1.0526 per single bet

The Expected net winnings in case of a black outcome for the single bet is $9.47.

It is given that

Number of black-colored​ balls= 18

Number of red-colored​ balls =18

Number of green-colored​ balls = 2

Total balls = 38

What is probability?

Probability is to quantify the possibilities or chances.

The probability of a black outcome will be:

Number of black-colored balls/total balls

Probability of a black outcome = 18/38

Similarly, the probability of a red outcome = 18/38

Probability of a green outcome = 2/38

if someone makes a $20 bet on black,

Expected net winnings = probability of black outcome * 20

i.e. (18/38) * 20

Expected net winnings = $9.47 for single bet

Therefore, the Expected net winnings in case of a black outcome for the single bet is $9.47.

To get more about probability visit:

https://brainly.com/question/25870256