Average Expected Outcome?

[afk]

A deck of 30 cards consisting of: 27 blanks, 2 $100 prizes, and 1 $500 prize, are arranged randomly except that each prize card is no less than 8 and no more than 13 cards away from another prize card. Each time a card is drawn it is discarded from play and the remaining cards maintain the same order until the player decides to end the game at which point all 30 cards are returned to the deck and randomized save for the rule of 8-13 above.

It costs $30 each time you draw a card.

What is the expected outcome of each new game?

 

[Deleted member 4993]

A deck of 30 cards consisting of: 27 blanks, 2 $100 prizes, and 1 $500 prize, are arranged randomly except that each prize card is no less than 8 and no more than 13 cards away from another prize card. Each time a card is drawn it is discarded from play and the remaining cards maintain the same order until the player decides to end the game at which point all 30 cards are returned to the deck and randomized save for the rule of 8-13 above.

It costs $30 each time you draw a card.

What is the expected outcome of each new game?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[afk]

As far as I have gotten is taking the best and worst case scenarios which are, +$470 and -$200 respectively, which are both equally probable and have come up with the answer that the average net expectation per new game is +$270, which is $470-$200.

The other answer that I believe may be correct is +$135, which is from adding the best and worst possible outlier scenarios then averaging them, (470 + -200)/2.

Are either of these answers correct?

Are they both wrong?

Why? Why not?

If an exact answer is too difficult to calculate can someone at least verify that the expectation is positive or negative?

And ultimately, given the information above, should the player play or not play the game if gaining money is the objective?

Your input is greatly appreciated!

 

[blamocur]

As far as I have gotten is taking the best and worst case scenarios which are, +$470 and -$200 respectively, which are both equally probable and have come up with the answer that the average net expectation per new game is +$270, which is $470-$200.

The other answer that I believe may be correct is +$135, which is from adding the best and worst possible outlier scenarios then averaging them, (470 + -200)/2.

Are either of these answers correct?

Are they both wrong?

Why? Why not?

If an exact answer is too difficult to calculate can someone at least verify that the expectation is positive or negative?

And ultimately, given the information above, should the player play or not play the game if gaining money is the objective?

Your input is greatly appreciated!

I am assuming from your post that the player keeps playing until she gets the $500 prize, right? Otherwise the statement "...until the player decides to end the game..." makes this problem undefined.

 

[afk]

I am assuming from your post that the player keeps playing until she gets the $500 prize, right? Otherwise the statement "...until the player decides to end the game..." makes this problem undefined.

As far as I have determined, yes, the player should always play until hitting the $500 and should always quit immediately after hitting it. However, the goal here is to win as much money as possible, so if for some reason there's any reason to continue after hitting the $500, assume the player does in order to maximize winnings, but I cannot see any way that this would be true... so refer to my first sentence in this reply please.

 

[BigBeachBanana]

I am assuming from your post that the player keeps playing until she gets the $500 prize, right? Otherwise the statement "...until the player decides to end the game..." makes this problem undefined.

Or we need to know the probability that the player will continue playing given how many prize cards and which one had been drawn. To simplify, we can model this by the amount of prize money is left in the deck.
[imath]\Pr(\text{Continue}|X=x)[/imath]; where [imath]x=\{0,100,200,500,600,700\}[/imath]
For example, [imath]\Pr(\text{Continue}|X=700)[/imath] is the probability that the player will continue to draw, given that there are still $700 left in the deck, i.e. no prize card had been drawn.

 

[afk]

Or we need to know the probability that the player will continue playing given how many prize cards and which one had been drawn. To simplify, we can model this by the amount of prize money is left in the deck.
[imath]\Pr(\text{Continue}|X=x)[/imath]; where [imath]x=\{0,100,200,500,600,700\}[/imath]
For example, [imath]\Pr(\text{Continue}|X=700)[/imath] is the probability that the player will continue to draw, given that there are still $700 left in the deck, i.e. no prize card had been drawn.

See above reply please! Given that once a winning card has been drawn, the minimum number of cards until the next winner is 8, and given that you can only win $100 or $500, once you hit the $500 there should never be a reason to continue as it will cost you $240 minimum to hit one of the remaining $100 winners (if any).

 

[afk]

Or we need to know the probability that the player will continue playing given how many prize cards and which one had been drawn. To simplify, we can model this by the amount of prize money is left in the deck.
[imath]\Pr(\text{Continue}|X=x)[/imath]; where [imath]x=\{0,100,200,500,600,700\}[/imath]
For example, [imath]\Pr(\text{Continue}|X=700)[/imath] is the probability that the player will continue to draw, given that there are still $700 left in the deck, i.e. no prize card had been drawn.

Also, using your formula, I believe this would be the correct condition to maximize winnings:
Pr(Continue∣X>=500)
Which equates to never continuing after drawing the $500 winning card.
But I'm not sure how to get to an actual answer with your formula?

 

[blamocur]

For this scenario my script tells me that the expected gain is $135, but I am not sure how to compute this by hand without coding.

 

[afk]

For this scenario my script tells me that the expected gain is $135, but I am not sure how to compute this by hand without coding.

Could you please share this script? Or at least explain what it is doing so that I can reproduce it on my end if you don't want to share the code for some reason?

 

[blamocur]

Could you please share this script? Or at least explain what it is doing so that I can reproduce it on my end if you don't want to share the code for some reason?

Please let me know if you find any bugs.
Thanks.

#!/usr/bin/env python3

sum = 0; ## Sum of gains for each configuration.
ctr = 0; ## Total number of configurations.

for gap3 in range (8, 14):                    # Gap between positions of 2nd and 3rd prize
    for gap2 in range (8, 14):                # Gap between positions of 1st and 2nd prize
        for gap1 in range (1 + 27-gap2-gap3): # number of cards before the first prize
            p1 = gap1;            # Position of the first prize
            p2 = p1 + 1 + gap2;   # Position of the first prize
            p3 = p2 + 1 + gap3;   # Position of the first prize

            winning1 = 500;             ## When the $500 is at p1
            winning2 = 100 + 500;       ## When the $500 is at p2
            winning3 = 100 + 100 + 500; ## When the $500 is at p3
            loss1    = (1+p1) * 30; ## When the $500 is at p1
            loss2    = (1+p2) * 30; ## When the $500 is at p1
            loss3    = (1+p3) * 30; ## When the $500 is at p1

            sum += (winning1-loss1);
            sum += (winning2-loss2);
            sum += (winning3-loss3);
            ctr += 3;
            if False:
                print ("%2d %2d %2d ... %2d %2d %2d" % (p1, p2, p3, gap1,gap2, gap3));

print ("Expected gain.:  %f" % (sum/ctr));