what is the expected value of the game to the​ player? if you played the game 1000​ times

Fred's Fun Manufactory

Task

A famous arcade in a seaside resort town consists of many dissimilar games of skill and chance. In order to play a popular “spinning wheel” game at Fred's Fun Mill Arcade, a player is required to pay a small, fixed corporeality of 25 cents each time he/she wants to make the cycle spin. When the wheel stops, the player is awarded tickets based on where the wheel stops -- and these tickets are and so redeemable for prizes at a redemption eye within the arcade.

Note: this particular game has no skill component; each spin of the wheel is a random event, and the result from each spin of the wheel is independent of the results of previous spins.

The wheel awards tickets with the post-obit probabilities:

ane ticket	35%
2 tickets	20%
3 tickets	20%
5 tickets	ten%
10 tickets	10%
25 tickets	iv%
100 tickets	1%

(Note: A motion-picture show of a wheel fitting these parameters is included.)

1. If a player were to play this game many, many times, what is the expected (average) number of tickets that the histrion would win from each spin?
2. The arcade often provides quarters to its customers in $5.00 rolls. Every day over the summertime, a young male child obtains i of these quarter rolls and uses all of the quarters for the spinning bicycle game. In the long run, what is the average number of tickets that this boy tin can wait to win each day using this strategy?
3. One of the redemption center prizes that the young boy is playing for is a trendy item that costs 300 tickets. It is as well available at a store down the street for $4.99. Without factoring in any enjoyment gained from playing the game or from visiting the arcade, from a strictly budgetary point of view, would y'all propose the boy to try and obtain this detail based on arcade ticket winnings or to just go and buy the item at the store? Explicate.

4. The histogram below summarizes the results of xc summer days of the boy playing the game using his "$5 roll per day" strategy. The first bar in the histogram represents those days where 40 to 59 tickets were won.

   
   1. For approximately what percent of the 90 days did the boy earn fewer than 100 tickets in a twenty-four hour period?
   2. For approximately what percentage of the xc days did the boy earn 200 or more than tickets in a day?
   3. For approximately what pct of the 90 days did the boy earn 300 or more tickets in a day?
5. To maintain the spinning wheel game machine, the arcade manager adopts a strategy of elimination the money box (that's where the quarters go after they are inserted in the machine) each time she refills the machine with a new roll of tickets. The ticket refill rolls contain 5000 tickets, and the machine is designed to hold $1300 in quarters in its money box. Presume that the machine was fully loaded with 5000 tickets and had an empty money box when it was outset used. Using the manager's maintenance strategy, is at that place any gamble that the money box could get completely full with quarters or overflow with quarters? Explain.

Arcade Wheel

Color	Tickets
Orangish	i
Light-green	two
Blue	3
Yellow	v
Imperial	10
Black	25
Cherry-red	100

Graphics Notation: The wheel diagram was developed in Microsoft Excel using its "Pie Chart" graph building feature. The intent is that each wedge represents 1% of the pie (3.half-dozen degrees). At that place are 35 orange wedges (each representing a win of 1 ticket) to correspond to the 35% probability of obtaining 1 ticket in a spin, xx greenish wedges (each representing a win of 2 tickets) to correspond to the 20% probability of obtaining 2 tickets in a spin, and then on.

IM Commentary

Developing the probability distribution of $X =$ the number of tickets earned over a set up of xx plays is an extremely daunting exercise. Yet, determining the average number of tickets earned over a gear up of twenty plays (in this case $E(ten)$) is accessible and can be approached using the formulas and concepts of expected value. Students tin can so use this knowledge to apply decision making in context; and via a simulation/sample of 90 cases, students can also have a fairly respectable sense as to the general commonness and rarity of possible values for $X$.

Solution

1. $ane \cdot .35 + two\cdot.2 + 3\cdot.2 + five\cdot.1 + 10\cdot.1 + 25\cdot.04 + 100\cdot.01 = 4.85 \text{ tickets}$. Note: If for some reason a student wanted to develop a simulation of several game plays to approximate this expected value (as opposed to using the formula), the student should simulate a number of spins, and the average number of tickets won per game should be very close to this number.
2. \$5 ringlet of quarters = 20 quarters, so $20 \cdot 4.85 = 97$ tickets each day, i.e., each prepare of 20 spins.
3. Various answers are possible, merely they should all deal with the thought that based on an expected average of 4.85 tickets for each game play, we expect it will crave most 62 game plays (61.86) on average to earn 300 tickets. 62 games * 25 cents = \$15.50, more iii every bit much as buying at the store. Other less formal arguments could use estimates or values obtained from questions (a) and (b) to a higher place. For instance, if the average is near 5 tickets per spin, a person needs about threescore spins on average to hit 300 tickets. 60 spins is \$fifteen which is three times more \$four.99, and then on. Or, with an boilerplate of almost 100 tickets a day (from question (b)), that means that iii days would be needed on boilerplate. That's three rolls of quarters = \$xv to get the 300 tickets, etc.
4. 1. $\frac{lx}{90} = 66.7$% (close approximations are acceptable, simply the histogram is designed such that lx cases fit the criteria out of a total of 90 cases).
   2. $\frac{6}{90} = 6.7$% (see note above)
   3. $\frac{0}{90} = 0$%
5. No. Various arguments can be made, but the central theme is that even if the game returns the minimum of only one ticket per play over 5000 plays (extremely unlikely), that represents only 5000 quarters = \$1250 which is less than the \$1300 chapters. Or from another perspective, the money box tin can hold 5200 quarters ($\frac{1300}{.25} = 5200$), and the about quarters that the auto can earn from a 5000 ticket roll would be 5000 quarters (once again, under the extremely unlikely minimal case of 5000 games of i ticket each).

Fred'south Fun Manufacturing plant

A famous arcade in a seaside resort town consists of many dissimilar games of skill and chance. In order to play a popular “spinning wheel” game at Fred'south Fun Factory Arcade, a player is required to pay a small-scale, fixed amount of 25 cents each time he/she wants to brand the bicycle spin. When the wheel stops, the actor is awarded tickets based on where the wheel stops -- and these tickets are then redeemable for prizes at a redemption center inside the arcade.

Annotation: this particular game has no skill component; each spin of the wheel is a random effect, and the issue from each spin of the wheel is independent of the results of previous spins.

The wheel awards tickets with the following probabilities:

1 ticket	35%
two tickets	20%
3 tickets	twenty%
5 tickets	10%
10 tickets	x%
25 tickets	iv%
100 tickets	1%

(Note: A picture of a cycle plumbing equipment these parameters is included.)

1. If a role player were to play this game many, many times, what is the expected (average) number of tickets that the player would win from each spin?
2. The arcade often provides quarters to its customers in $5.00 rolls. Every solar day over the summer, a young boy obtains ane of these quarter rolls and uses all of the quarters for the spinning bike game. In the long run, what is the boilerplate number of tickets that this boy can expect to win each day using this strategy?
3. One of the redemption center prizes that the immature boy is playing for is a trendy detail that costs 300 tickets. Information technology is also available at a store down the street for $4.99. Without factoring in whatsoever enjoyment gained from playing the game or from visiting the arcade, from a strictly budgetary point of view, would you advise the boy to endeavor and obtain this particular based on arcade ticket winnings or to just go and buy the item at the store? Explain.

4. The histogram beneath summarizes the results of 90 summer days of the boy playing the game using his "$5 coil per mean solar day" strategy. The starting time bar in the histogram represents those days where xl to 59 tickets were won.

   
   1. For approximately what percent of the xc days did the boy earn fewer than 100 tickets in a mean solar day?
   2. For approximately what per centum of the 90 days did the boy earn 200 or more than tickets in a day?
   3. For approximately what percentage of the 90 days did the boy earn 300 or more than tickets in a day?
5. To maintain the spinning wheel game machine, the arcade manager adopts a strategy of elimination the money box (that's where the quarters get after they are inserted in the machine) each fourth dimension she refills the car with a new curl of tickets. The ticket refill rolls contain 5000 tickets, and the car is designed to hold $1300 in quarters in its money box. Assume that the car was fully loaded with 5000 tickets and had an empty money box when it was commencement used. Using the director'due south maintenance strategy, is there any risk that the money box could become completely full with quarters or overflow with quarters? Explain.

Arcade Wheel

Color	Tickets
Orange	1
Light-green	2
Blue	3
Yellow	v
Purple	10
Black	25
Ruby	100

Graphics Note: The bicycle diagram was developed in Microsoft Excel using its "Pie Chart" graph edifice feature. The intent is that each wedge represents one% of the pie (three.half-dozen degrees). There are 35 orange wedges (each representing a win of 1 ticket) to correspond to the 35% probability of obtaining 1 ticket in a spin, twenty green wedges (each representing a win of 2 tickets) to correspond to the 20% probability of obtaining 2 tickets in a spin, and so on.

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Source: https://tasks.illustrativemathematics.org/content-standards/tasks/1197

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