a) All the values are equally likely, so to get the expected value of the die we just sum all possible values and divide by the total amount of possible values, 6.

$\frac{1+2+3+4+5+6}{6} = 3.5$

Thus, the expected value of a die is 3.5.

b) The fair value is the expected value of the game. You will reroll the dice only if you expect to get more value on the second throw than what did you obtain on the first one. Since the expected value of a die is 3.5, then you will only reroll the die if you obtain 3 or less, and the probability for that is 1/2.

This means that there is a probability of 1/2 to make a second roll, with expected value 3.5, and a probability of 1/6 to obtain each of 4, 5 or 6 with the first die and keep that value (because it is higher than the expected value from one dice). Therefore, the expected value of this game is

1/2 * 3.5 + 1/6 (4+5+6) = 4.25

The fair value of the game is 4.25 dollars.

c) You can interpret this game as if after your first roll, you can keep that value or play the game described in b, where you roll once and reroll once if you want. Thus, if you decide to decline the value of the first die and reroll, then the expected value of the game will be 4.25. As a result, you will only reroll if you obtain a 4 or lower in the first die, which happens with a probability of 4/6 = 2/3. If you obtain 5 or 6, which happen each case with probability 1/6, you will keep that value because it exceeds your expectations. As a consecuence, the expected value is

2/3 * 4.25 + 1/6*(5+6) = 14/3 = 4.6667

The fair value of the game is 14/3 = 4.6667 dollars.

7 0

3 years ago

The mean of four numbers is 70. When a fifth number is included, the mean of the five numbers is 80. What is the fifth number?

Yakvenalex [24]

<h3>Answer: 120</h3>

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Explanation:

S = sum of the first four values

S/4 = 70 since we divide the sum by 4 to get the mean of the first four values

S = 280 is the result after multiplying both sides by 4

Let x be the fifth unknown value. Adding it onto S leads to S+x = 280+x. This represents the sum of all five values.

Dividing that sum 280+x over 5 leads to the new mean 80, which leads to...

----------

(280+x)/5 = 80

280+x = 5*80

x+280 = 400

x = 400-280

x = 120

The fifth number is 120

3 0

3 years ago

Kate has a rectangular garden that is 8 feet long by 3 feet wide. She wishes to extend the length and width by the same amount i

elena55 [62]

<span>Let the extension of the length and width be x.
The new dimensions are: Length = 8+x
Width =3+x
Area = 104 ft squared
Area of a rectangle = Length x Width
Therefore,
(8+x)(3+x) = 104
x^2 +11x +24=104
x^2 +11x +24-104=0
x^2 +11x-80=0

Factorize using splitting the middle term method. X^2 +16x-5x-80=0 x(x+16)-5(x+16)=0 (x-5)(x+16)=0 either x-5 = 0 or x+16=0 x=5 or x=-16 length and width measure cannot be negative. Therefore, the extension of the length and width is 5 feet each.</span>

4 0

3 years ago