1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
d1i1m1o1n [39]
3 years ago
7

In a casino game, gamblers are allowed to roll n fair, 6-sided dice. If a 6 shows up on any of them, the gambler gets nothing. I

f no 6’s appear, the gambler is paid the sum of the values on the dice in dollars. The gambler is free to choose n, the number of dice rolls.
a) Derive a formula for the gambler’s expected payoff (the total dollars won). Plot this payoff for values of n from 1 to 20. What is the smallest n that maximizes the expected payoff ?

b) Suppose the gambler chooses to roll 10 dice (this is not necessarily the answer to part (a)). What is the expected number of distinct dice values that show up? In other words, what is the expected number of faces that are rolled at least once?

Mathematics
1 answer:
Soloha48 [4]3 years ago
4 0

Answer:

a) \bf \boxed{W(n) = 3*n*\left( \frac{5}{6} \right)^n\;dollars}

the smallest n that maximizes the expected payoff is n=5.

b) 4

Step-by-step explanation:

a)

The expected amount of $ won for each die would be the average of 1, 2, 3, 4 and 5 which is $3.

Let W(n) the expected money won when rolling n dice.  

n =1

If the gambler rolls only one die, the expected money won would be $3 times the probability of not getting a 6, which is 5/6.

So  

\bf W(1) = 3*1*\frac{5}{6}

n=2

If the gambler rolls 2 dice, the expected money won would be $3 times the probability of not getting a 6 in any of the dice. Since the outcome of the rolling does not depend on the previous rollings, the probability is  

\bf \frac{5}{6}\times\frac{5}{6}=\left( \frac{5}{6} \right)^2

and

\bf W(2) = 3*2*\left( \frac{5}{6} \right)^2

n=3

Similarly, since the probability of not getting a 6 in 3 dice equals

\bf \left( \frac{5}{6} \right)^3

\bf W(3) = 3*3*\left( \frac{5}{6} \right)^3

and the formula for the expected money won with n dice would be

\bf \boxed{W(n) = 3*n*\left( \frac{5}{6} \right)^n\;dollars}

In the picture attached there is a plot of the values of the expected money won for n=1 to 20 (See picture)

For n=5 and n=6 we get the maximum profit expected of $6.02816=$6 rounded to the nearest integer.

Hence, the smallest n that maximizes the expected payoff is n=5.

b)

The probability that face k (k=1,2,...or 6) shows up is 1/6,

as this face can be in any of the 10 positions of the arrangement, there are 10 ways that face k can show up.

The probability that face k (k=1,2,...or 6) shows up twice is \bf \left( \frac{1}{6} \right)^2

as this face can be in any of the \bf C(10;2)=\binom{10}{2} (combinations of 10 taken 2 at a time) positions of the arrangement, there are C(10;2) ways that face k can show up twice.

The probability that face k (k=1,2,...or 6) shows up three times is \bf \left( \frac{1}{6} \right)^3

as this face can be in any of the \bf C(10;3)=\binom{10}{2} (combinations of 10 taken 2 at a time) positions of the arrangement, there are C(10;3) ways that face k can show up twice.

So, we infer that the expectation is  

\bf \sum_{k=1}^{10}\binom{10}{k}(1/6)^k=3.6716\approx 4

and the expected number of distinct dice values that show up is 4.

You might be interested in
Please help multiple choice !!
professor190 [17]
The answer is x= 2, y= 1
8 0
2 years ago
Is 7/12 greater then 4/12
aev [14]
True.

7/12 is greater than 4/12.

Have a wonderful day! :)

6 0
2 years ago
Read 2 more answers
PLZZ HELP 20 POINTS!!
katen-ka-za [31]

Answer:

son 60 puntos ...!!!!!!!!

8 0
2 years ago
Read 2 more answers
Determine whether y varies directly with x. If so, find the constant of variation k.
musickatia [10]
The answer is the option D because the formula is y upon x
3 0
3 years ago
How to eat hotdog thank you
lana66690 [7]

Answer:

1.Toast your hot dog bun if you like.

2.Grab pan with little bit of water

4.Place hotdog and cook until chang color on both sides

5. Place a cooked hot dog inside the bun.

6.Add sauces.

6.Put on chunky condiments next.

7.Top it with cheese if you like.

8.Finish the dog with your favorite spices.

9. Put the assembled hot dog on a plate.

10.Serve your favorite snacks on the side.

6 0
2 years ago
Read 2 more answers
Other questions:
  • Change your money from dollars to british pounds.you got 560 pounds for 800 dollars.how many pounds will you get for 300 dollars
    15·1 answer
  • The owner of a video game store creates the expression −2x2 + 28x + 20 to represent the store's weekly profit in dollars, where
    6·1 answer
  • Please answer my fraction questions
    11·1 answer
  • Henry and Jose are gaining weight for football. Henry weighs 205 pounds and is gaining 2 pounds per week. Jose weighs 195 pounds
    6·1 answer
  • How many dimensions can a polygon grow from?
    10·1 answer
  • Consider 8x2 - 48x = -104. Write the equation so that a = 1: x2 + __ x = __
    5·1 answer
  • Which of the following is a required deduction? O A. Medicare B. Health insurance C. Disability insurance D. Medicaid
    10·1 answer
  • Which measurement is equivalent to 880 cm?
    11·1 answer
  • Find the standard form of the following...<br> 26
    14·1 answer
  • What is the slope AND y-intercept to this equation: y=6x-8
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!