iTunes-Pepsi Bottles Delayed

[Doctor Q]

I'm glad you asked, wdlove. I'm sure you wanted a math lesson, right?

The chances of getting 3 out of 6 with 1 in 3 odds can be computed with a formula (the clever and efficient way), or by enumeration (the brute force long-winded way).

To maximum the fun, I'll choose to solve it the long-winded way first!

Each of the 6 bottles has a 1 in 3 in chance of being a winner. So over a large number of samples, you'll have 2 losers for every winner. That means that these three situations are equally likely:

LOSE
LOSE
WIN

Let's let 0 stand for LOSE and 1 stand for WIN. So we can abbreviate the possibilities as follows: 001, meaning that, on average, you will get 0 (lose) twice and 1 (win) once in 3 tries.

With two bottles, these 3x3=9 cases are equally likely:

1stbottle 2ndbottle:

LOSE LOSE
LOSE LOSE
LOSE WIN

LOSE LOSE
LOSE LOSE
LOSE WIN

WIN LOSE
WIN LOS
WIN WIN

Abbreviate this to
00 00 01 00 00 01 10 10 11

So over the 9 tries you would average no winners 4 times, 1 winner 4 times, and two winners once.

For 3 bottles:
000 000 001 000 000 001 010 010 011
000 000 001 000 000 001 010 010 011
100 100 101 100 100 101 110 110 111

Over 27 tries, you get no winners 8 times, 1 winner 12 times, 2 winners 6 times, and 3 winners once.

For 4 bottles:
0000 0000 0001 0000 0000 0001 0010 0010 0011
0000 0000 0001 0000 0000 0001 0010 0010 0011
0100 0100 0101 0100 0100 0101 0110 0110 0111

0000 0000 0001 0000 0000 0001 0010 0010 0011
0000 0000 0001 0000 0000 0001 0010 0010 0011
0100 0100 0101 0100 0100 0101 0110 0110 0111

1000 1000 1001 1000 1000 1001 1010 1010 1011
1000 1000 1001 1000 1000 1001 1010 1010 1011
1100 1100 1101 1100 1100 1101 1110 1110 1111

For 5 bottles:
00000 00000 00001 00000 00000 00001 00010 00010 00011
00000 00000 00001 00000 00000 00001 00010 00010 00011
00100 00100 00101 00100 00100 00101 00110 00110 00111

00000 00000 00001 00000 00000 00001 00010 00010 00011
00000 00000 00001 00000 00000 00001 00010 00010 00011
00100 00100 00101 00100 00100 00101 00110 00110 00111

01000 01000 01001 01000 01000 01001 01010 01010 01011
01000 01000 01001 01000 01000 01001 01010 01010 01011
01100 01100 01101 01100 01100 01101 01110 01110 01111

00000 00000 00001 00000 00000 00001 00010 00010 00011
00000 00000 00001 00000 00000 00001 00010 00010 00011
00100 00100 00101 00100 00100 00101 00110 00110 00111

00000 00000 00001 00000 00000 00001 00010 00010 00011
00000 00000 00001 00000 00000 00001 00010 00010 00011
00100 00100 00101 00100 00100 00101 00110 00110 00111

01000 01000 01001 01000 01000 01001 01010 01010 01011
01000 01000 01001 01000 01000 01001 01010 01010 01011
01100 01100 01101 01100 01100 01101 01110 01110 01111

10000 10000 10001 10000 10000 10001 10010 10010 10011
10000 10000 10001 10000 10000 10001 10010 10010 10011
10100 10100 10101 10100 10100 10101 10110 10110 10111

10000 10000 10001 10000 10000 10001 10010 10010 10011
10000 10000 10001 10000 10000 10001 10010 10010 10011
10100 10100 10101 10100 10100 10101 10110 10110 10111

11000 11000 11001 11000 11000 11001 11010 11010 11011
11000 11000 11001 11000 11000 11001 11010 11010 11011
11100 11100 11101 11100 11100 11101 11110 11110 11111

For 6 bottles:
000000 000000 000001 000000 000000 000001 000010 000010 000011
000000 000000 000001 000000 000000 000001 000010 000010 000011
000100 000100 000101 000100 000100 000101 000110 000110 000111

000000 000000 000001 000000 000000 000001 000010 000010 000011
000000 000000 000001 000000 000000 000001 000010 000010 000011
000100 000100 000101 000100 000100 000101 000110 000110 000111

001000 001000 001001 001000 001000 001001 001010 001010 001011
001000 001000 001001 001000 001000 001001 001010 001010 001011
001100 001100 001101 001100 001100 001101 001110 001110 001111

000000 000000 000001 000000 000000 000001 000010 000010 000011
000000 000000 000001 000000 000000 000001 000010 000010 000011
000100 000100 000101 000100 000100 000101 000110 000110 000111

000000 000000 000001 000000 000000 000001 000010 000010 000011
000000 000000 000001 000000 000000 000001 000010 000010 000011
000100 000100 000101 000100 000100 000101 000110 000110 000111

001000 001000 001001 001000 001000 001001 001010 001010 001011
001000 001000 001001 001000 001000 001001 001010 001010 001011
001100 001100 001101 001100 001100 001101 001110 001110 001111

010000 010000 010001 010000 010000 010001 010010 010010 010011
010000 010000 010001 010000 010000 010001 010010 010010 010011
010100 010100 010101 010100 010100 010101 010110 010110 010111

010000 010000 010001 010000 010000 010001 010010 010010 010011
010000 010000 010001 010000 010000 010001 010010 010010 010011
010100 010100 010101 010100 010100 010101 010110 010110 010111

011000 011000 011001 011000 011000 011001 011010 011010 011011
011000 011000 011001 011000 011000 011001 011010 011010 011011
011100 011100 011101 011100 011100 011101 011110 011110 011111

000000 000000 000001 000000 000000 000001 000010 000010 000011
000000 000000 000001 000000 000000 000001 000010 000010 000011
000100 000100 000101 000100 000100 000101 000110 000110 000111

000000 000000 000001 000000 000000 000001 000010 000010 000011
000000 000000 000001 000000 000000 000001 000010 000010 000011
000100 000100 000101 000100 000100 000101 000110 000110 000111

001000 001000 001001 001000 001000 001001 001010 001010 001011
001000 001000 001001 001000 001000 001001 001010 001010 001011
001100 001100 001101 001100 001100 001101 001110 001110 001111

000000 000000 000001 000000 000000 000001 000010 000010 000011
000000 000000 000001 000000 000000 000001 000010 000010 000011
000100 000100 000101 000100 000100 000101 000110 000110 000111

000000 000000 000001 000000 000000 000001 000010 000010 000011
000000 000000 000001 000000 000000 000001 000010 000010 000011
000100 000100 000101 000100 000100 000101 000110 000110 000111

001000 001000 001001 001000 001000 001001 001010 001010 001011
001000 001000 001001 001000 001000 001001 001010 001010 001011
001100 001100 001101 001100 001100 001101 001110 001110 001111

010000 010000 010001 010000 010000 010001 010010 010010 010011
010000 010000 010001 010000 010000 010001 010010 010010 010011
010100 010100 010101 010100 010100 010101 010110 010110 010111

010000 010000 010001 010000 010000 010001 010010 010010 010011
010000 010000 010001 010000 010000 010001 010010 010010 010011
010100 010100 010101 010100 010100 010101 010110 010110 010111

011000 011000 011001 011000 011000 011001 011010 011010 011011
011000 011000 011001 011000 011000 011001 011010 011010 011011
011100 011100 011101 011100 011100 011101 011110 011110 011111

100000 100000 100001 100000 100000 100001 100010 100010 100011
100000 100000 100001 100000 100000 100001 100010 100010 100011
100100 100100 100101 100100 100100 100101 100110 100110 100111

100000 100000 100001 100000 100000 100001 100010 100010 100011
100000 100000 100001 100000 100000 100001 100010 100010 100011
100100 100100 100101 100100 100100 100101 100110 100110 100111

101000 101000 101001 101000 101000 101001 101010 101010 101011
101000 101000 101001 101000 101000 101001 101010 101010 101011
101100 101100 101101 101100 101100 101101 101110 101110 101111

100000 100000 100001 100000 100000 100001 100010 100010 100011
100000 100000 100001 100000 100000 100001 100010 100010 100011
100100 100100 100101 100100 100100 100101 100110 100110 100111

100000 100000 100001 100000 100000 100001 100010 100010 100011
100000 100000 100001 100000 100000 100001 100010 100010 100011
100100 100100 100101 100100 100100 100101 100110 100110 100111

101000 101000 101001 101000 101000 101001 101010 101010 101011
101000 101000 101001 101000 101000 101001 101010 101010 101011
101100 101100 101101 101100 101100 101101 101110 101110 101111

110000 110000 110001 110000 110000 110001 110010 110010 110011
110000 110000 110001 110000 110000 110001 110010 110010 110011
110100 110100 110101 110100 110100 110101 110110 110110 110111

110000 110000 110001 110000 110000 110001 110010 110010 110011
110000 110000 110001 110000 110000 110001 110010 110010 110011
110100 110100 110101 110100 110100 110101 110110 110110 110111

111000 111000 111001 111000 111000 111001 111010 111010 111011
111000 111000 111001 111000 111000 111001 111010 111010 111011
111100 111100 111101 111100 111100 111101 111110 111110 111111

If you scan these patterns, you'll see that there are 20 patterns that give you three wins (111000, 110100, 110010, 110001, 101100, 101010, 101001, 100110, 100101, 100011, 011100, 011010, 011001, 010110, 010101, 010011, 001110, 001101, 001011, 000111) and each occurs 8 times, so we get a total of 20x8=160 ways to win out of 3x3x3x3x3x3=729 tries. So the odds of getting 3 wins with 6 bottles is 160/729 = 21.95%.

--- Pause for a Pepsi, then on to my next post ---

 

[Doctor Q]

OK, now we'll try it with formulas!

Suppose you open 6 bottles, one at a time, and the first 3 turn out to be winners and the next 3 turn out to be losers. The odds of the first bottle being a winner was 1/3. The second, 1/3. The third, 1/3. The odds of the fourth bottle being a loser was 2/3. The fifth, 2/3. The sixth, 2/3. The combined odds are therefore 1/3 x 1/3 x 1/3 x 2/3 x 2/3 x 2/3 = (1/3)^3 x (2/3)^3 = 8/729.

But getting 3 winners and then 3 losers is only one of the ways to get 3 winners out of 6, so we have to multiply our result by the number of orders in which you can win 3 out of 6. This is called the number of combinations of 3 out of 6, written C(6,3). The formula is 6! divided by 3! x (6-3)!, or 6! / (3! x 3!) = 20.

The final answer: Our odds are 8/729 x 20 = 160/729 = 21.95%

In general, the chances of getting R winning bottles out of N bottles you buy (without peeking under the cap) is:

N!
--------- x (1/3)^R x (2/3)^(N-R)
R! (N-R!)


0 out of 6: 64/729 = 8.78%
1 out of 6: 192/729 = 26.34%
2 out of 6: 240/729 = 32.92%
3 out of 6: 160/729 = 21.95%
4 out of 6: 80/729 = 8.23%
5 out of 6: 12/729 = 1.65%
6 out of 6: 1/729 = 0.14%
(The total isn't exactly 100% due to rounding.)

Finally, you can sum these odds to find the odds of getting at least R winning bottles out of N bottles:

at least 0 out of 6: 729/729 = 100%
at least 1 out of 6: 665/729 = 91.22%
at least 2 out of 6: 473/729 = 64.88%
at least 3 out of 6: 253/729 = 34.71%
at least 4 out of 6: 93/729 = 12.76%
at least 5 out of 6: 13/729 = 1.78%
at least 6 out of 6: 1/729 = 0.14%

Does that answer your question?

 

[wdlove]

Very interesting indeed Doctor Q! Yes, it more than answers my question. I appreciate you being so thorough. My guess is that you have a love for math.

 

[ddbean]

... Diet Pepsi iTunes bottles at Smart & Final in Ventura County, .... I'm not going to tilt & peek, just open at random and see what my tally is. ....

Here's my tally for a 24 case.
12 winners
12 play again

Glad to see it was better than 3-1 odds for me. Guess that means somewhere someones getting rotten odds, uh?

 

[Doctor Q]

Finally!

Yesterday, I spotted a 1-liter PepsiTunes bottle for the first time, at a Ralphs supermarket in Los Angeles.

 

[Applexilef]

They are still no where to be found in Orlando... Though Sam's has them in the 12oz bottles. 7 11 does have the special big gulp gulps... I've been drinking a lot of em... full of coke though

Where in orlando? I'm next to UCF and I see them all the time, and in all types (PEPSI, Sierra Mist, 7-11)

 

[wdlove]

Yesterday, I spotted a 1-liter PepsiTunes bottle for the first time, at a Ralphs supermarket in Los Angeles.

I still haven't noticed any 1-liter Pepsi. Did you purchase and was it a winner Doctor Q? I didn't go to my job this week because of my news. Do wonder if they will still have any yellow caps this week.

 

[FriarTuck]

I'm 4 for 4 on the 1 liter Diet Pepsis.

And 1 for 4 on 20 ouncers.

I'll be getting liters from here until the end of the month.

 

[Doctor Q]

I still haven't noticed any 1-liter Pepsi. Did you purchase and was it a winner Doctor Q?

No and I don't know. I never buy that size. A can or a 20oz bottle can be a one-person serving, but soda gets flat too fast for me to use up a 1 liter bottle. And if I'm having a party, I'd buy 2-liter bottles.

 

[Urdam]

This is Pepsi's fault

 

[JohnGillilan]

I won 27 out of a batch of 50. Not too shabby.

 

[wickster]

AK in good supply

I work at a gas station and we have been getting steady shipments of 20oz Sierra Mist, Pepsi, and Diet Pepsi also 1 Liter Pepsi and Diet Pepsi. I live in rural Alaska with a population of 1500 people. Not sure why ya'll dont have access to iTunes Pepsi bottles.

 

[JGowan]

To you, sir, I say DUH.

...you can read the two line code. So hell i don't even buy the bottles. The people in my town don't even have computers so i'm doing Apple a favor and getting them some redemption. I'm 41/41 so far. Try it just to check me I don't lie.

A gas station attendant is aware that NO ONE in his town owns a computer. Right.

Great. People buy a winning bottle and you've stolen their song. And your assumption that "people in my town don't even have computers" is just your conscience trying to feel better about stealing all around: stealing from Pepsi and not buying their soda and then, stealing from the people who legitmately should have the song.

And the glee with which you posted just screams that you have no idea that what you've done is underhanded and wrong. If you did the "tilt" thing and then bought the drinks that had free songs, I could understand more. At least your taking the bottles out of the market.

You make Pepsi and Apple look terrible. People wil buy the bottles and go to collect and find that their code is no good. No one will suspect that a jerk at their store is stealing from them. It would be bad if you said that you were just a customer going in an writing down codes, but as a worker, it is really awful. I can just see you after work, locking the door and heading for the back, going through case after case with your little clipboard and then boxing everything back up and feeling so smug. I hope there are cameras that you're unaware of and that your boss figures out what you're doing and cans your ASS. Then you can tell your family, "Oh, I thought $40 worth of inferior encoded songs was worth more than having a steady paycheck."

----

FYI, MacRumors: Wickster came back on and edited out the part about his stealing the 41 tunes just by reading the codes on the bottles where he works and leaving the bottles without paying for them. Trust me, he said it.