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notka56 [123]
3 years ago
12

The game of european roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. a ball is spun onto the wh

eel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. gamblers can place bets on red or black. if the ball lands on their color, they double their money. if it lands on another color, they lose their money.
Mathematics
2 answers:
Goryan [66]3 years ago
6 0

Answer:

a) Expected amount of winnings = -$0.084

Standard deviation = $3.00

b) Expected amount of winnings = -$0.084

Standard deviation = $1.73

c) The expected winnings for both cases is exactly the same, but the spread of the wins about the expected value/mean is low in the case (b) compared to case (a).

So, with the expected winnings using the method in (a) and (b) the same, the risk associated with betting $3 dollar per game is higher because it has a higher standard deviation. This points to higher spread of losses and wins betting $3 once than betting $1 three different times.

Step-by-step explanation:

(a) Suppose you play roulette and bet $3 on a single round. What is the expected value and standard deviation of your total winnings?

It's either one wins or not with a bet of $3.

We can then obtain the probability mass function.

Random variable X represents the amount of winnings.

- If one bets $3 and picks the right colour, the person wins $6.

X = amount of winnings = 6 - 3 = $3

probability of winning = (18/37) = 0.486

- If one bets $3 and picks the wrong colour, then the person wins $0.

X = amount of winnings = 0 - 3 = -$3

Probability of losing = (19/37) = 0.514

The probability mass function is then

X | P(X)

3 | 0.486

-3 | 0.514

E(X) = Σ xᵢpᵢ

xᵢ = each variable

pᵢ = probability of each variable

E(X) = (3×0.486) + (-3×0.514) = -$0.084

Standard deviation = √(variance)

Variance = Var(X) = Σx²p − μ²

μ = E(X) = -0.084

Σx²p = (3²×0.486) + [(-3)² × 0.514] = 9

Variance = 9 - (-0.084)² = 8.993

Standard deviation = √8.993 = 2.999 = $3

(b) Suppose you bet $1 in three different rounds. What is the expected value and standard deviation of your total winnings?

With a bet of $1,

It's either one wins or not with a bet of $1.

We can then obtain the probability mass function.

Random variable X represents the amount of winnings.

- If one bets $1 and picks the right colour, the person wins $2.

X = amount of winnings = 2 - 1 = $1

probability of winning = (18/37) = 0.486

- If one bets $1 and picks the wrong colour, then the person wins $0.

X = amount of winnings = 0 - 1 = -$1

Probability of losing = (19/37) = 0.514

The probability mass function is then

X | P(X)

1 | 0.486

-1 | 0.514

E(X) = Σ xᵢpᵢ

xᵢ = each variable

pᵢ = probability of each variable

E(X) = (1×0.486) + (-1×0.514) = -$0.028

For 3 rounds of $1, the expected amount of winnings = 3 × -0.028 = -$0.084

Variance = Var(X) = Σx²p − μ²

μ = E(X) = -0.028

Σx²p = (1²×0.486) + [(-1)² × 0.514] = 1

Variance = 1 - (-0.028)² = 0.999216

Variance for combining 3 independent distributions = Σ λᵢ²σᵢ²

Variance of 3 rounds of $1 bets = 3[1²× var(X)] = 3 × var(X) = 3 × 0.999216 = 2.997648

Standard deviation = √(variance) = √(2.997648) = $1.73

(c) How do your answers to parts (a) and (b) compare? What does this say about the riskiness of the two games?

The expected winnings for both cases is exactly the same, but the spread of the wins about the expected value/mean is low in the case (b) compared to case (a).

So, with the expected winnings using the method in (a) and (b) the same, the risk associated with betting $3 dollar per game is higher because it has a higher standard deviation. This points to higher spread of losses and wins betting $3 once than betting $1 three different times.

Hope this Helps!!!

Dafna11 [192]3 years ago
5 0
R u supposed to multiply this or divide
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Answer:

B) Multiplying by 0

Step-by-step explanation:

This answer is correct because lets say we have a nonzero number, (IN this case lets say 4) 4+0 is 4, and that does not change the value because it is still 4, so A is not correct. Multiplying by 1 and dividing by one leads to the same conclusion. They both wind up to 4. The only one is B) because 4x0 is 0 and that isn't 4, so it changes value.

Hope this helped :)

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Answer:

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Step-by-step explanation:

64/80 = .8/1

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7 0
3 years ago
21 Maria sold small boxes of candy for $1 and
Anna35 [415]

Answer:

There where 10 <u>more</u> Small boxes sold than Large boxes.

Step-by-step explanation:

<u>This is a typical question that can be solved by obtaining a system of two linear equations and solving for each variable. </u>

In principle Linear Equations are algebraic expressions denoting a relationship between a Dependent variable y and an Independent variable x. In a system of Two Linear equations we have<u> two equations</u> of the same variable sets (thus Two Independent variables) so in this case both y and x will be variable terms.

Now with respect to the question and the given information, here our two Variable terms will be the small and the large boxes.

<u>Given Information:</u>

  • Small Boxes (lets call them s) cost $1 per box
  • Large Boxes (lets call them l) cost $4 per box
  • Total Number of Boxes sold is 30
  • Total Profit from sold Boxes is $60

Thus from the above we can obtain one equation denoting the Total Number of Boxes sold and one equation denoting the Total Profit from sold boxes, respectively, as follow:

s+l=30        Eqn(1): Total Number of Boxes

1s+4l=60     Eqn(2): Total Profit from sold boxes

Now we have a system of two linear equations which we can solve and find the number of small and large boxes, s and l respectively.

From Eqn(1) we see that

s=30-l      Eqn(3).

Plugging Eqn(3) in Eqn(2) we can solve for l as:

1(30-l)+4l=60

30-l+4l=60       Factored out bracket

3l=60-30          Gather similar terms on each side and simplify

3l=30

l=\frac{30}{3}

l=10

Plugging in the value for l=10 in Eqn(3) we have

s=30-10\\s=20

So we know that they were<u> 10 Large Boxes and 20 Small Boxes sold</u>, thus to answer our question, there where 10 more Small boxes sold than Large boxes.

5 0
3 years ago
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