Winter event: what are the odds?

[NandodeMC]

Today the winter train selection kit came out as daily prize on live and I went to spend my stars on the event board. So I started opening presents; one among the first ones was the “Show 2”, which revealed the “Shuffle” and other prize, but I obviously ignored them both.

I kept on opening presents but the train selection kit didn’t show up... and it didn’t just until the very last one! I made quick calculations about the odds of this happening but the result was quite absurd, and I just don’t want to believe it...

Can anyone tell what are the actual odds? Thanks beforehand and Merry Christmas!

 

[xivarmy]

Today the winter train selection kit came out as daily prize on live and I went to spend my stars on the event board. So I started opening presents; one among the first ones was the “Show 2”, which revealed the “Shuffle” and other prize, but I obviously ignored them both.

I kept on opening presents but the train selection kit didn’t show up... and it didn’t just until the very last one! I made quick calculations about the odds of this happening but the result was quite absurd, and I just don’t want to believe it...

Can anyone tell what are the actual odds? Thanks beforehand and Merry Christmas!

The odds of the train selection kit having been the last one you check are simply 1/18 (=5.6%) because there's 18 presents - you would normally not see how often this happens as you usually simply stumble into the shuffle first unless the show 2 shows it to you - and whether the train kit would've been the next one you open or last one you open after that is irrelevant.

The show 2 itself is just 2 additional reveals safely and does not change the odds.

 

[CrashBoom]

100%

it will sure happen to some players

The odds of the train selection kit having been the last one you check are simply 1/18 (=5.6%) because there's 18 presents

he already opened the see 2 (and a few other: "one among the first ones" so not really the 1st of all)
and is seeing 2

so let's say 10-15 unknown

1/10= 10%
1/15 = 6,67%

that are the odds of getting it as last AFTER already opening the show 2 in the first ones (1-5)

The odds of the train selection kit having been the last one you check are simply 1/18 (=5.6%) because there's 18 presents

you can't check the last one (18th) if it is not the shuffle

 

[Sl8yer]

50%

It happens, or it doesn't happen.

 

[Darrth Eugene Vader3]

50%
It happens, or it doesn't happen.

The probability of next present opened is the Winter Train is not 50%. This is not like tossing a perfectly balanced coin where there are:

• only two possible outcomes (ruling out that the coin would end up standing straight, not laying on one side head/tail); and
• each possible outcomes has the same odds to happen.

In this case we have 18 presents to open (at first attempt) and only one is the Winter Train.
P1(WIN) = Probability (It happens, at 1st attempt) = Number of Gifts with the Winter Train / Total number of Gifts available to choose. = 1/18 = 5.56%
P1(FAIL) =Probability (It does not happen, 1st attempt) = Number of Gifts not Winter Train / Total number of Gifts available to choose. = 17/18 = 94.44%
===
After that 1st attempt, and if the 1st attempt is not the Winter Train:

• If the Shuffle was obtained, then we get a new set of 18 closed presents,. so we got again a 1st attempt with 1/18 probability of getting the desired Winter Train at next attempt. P2(WIN) = a new P1(WIN) = 1/18 = 5.556% P2(FAIL) = 94.44%
• If the " Show Two" was obtained, and the two presents shown include the Winter Train, then the probability of getting the desired Winter Train at next attempt is : P2(WIN) = 100% (Winter Train location is revealed by the Show Two). P2(FAIL) =0%
• If the " Show Two" was obtained, and the two presents shown do not include the Winter Train, then the number of ways to get the Winter Train at next attempt are changed to 1 out of 15 (there are now 17 closed presents, 2 presents are revealed as not the Winter Train, 1 of the remaining 15 is the Winter Train). P2(WIN) in this scenario is: P2(WIN) = 1/15 = 6.67%, P2(FAIL) = 1 - P2(WIN) = 93.33%
• If the Show Two was not obtained, then we get 17 presents to open and just 1 of them is the Winter Train, P2(WIN) is in this case: 1/17 or 5.88%,
  P2(FAIL) = 1 - P2(WIN) = 94.12%

===
As the Player continues, and if the Shuffle is not obtained, the odds to get the Winter Train increase since there is at each turn one less closed/unknown gifts. Next odds would be 1/16 or 1/14 depending if the Show Two item is obtained at the second attempt or not.
===
Edit:
To be 50% It happens, or it doesn't happen we would need at first attempt: 9 Gifts with Winter Train, and 9 gifts with anything else.
or
To be 50% It happens or it doesn't happen, the Player must have opened 16 gifts and the last two remaining be the Winter Train and the Shuffle, both of them closed, none revealed by the Show Two gift. Then, and only then, the probability of next present opened is the Winter Train is 50%.

Source: Own knowledge, Probability & Statistics, university course, August - December 1990.
Google Topic: Conditional Probability
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred.

 

[Darrth Eugene Vader3]

Odds on getting the Winter Train, starting with the following events already occurred:

• Show Two already obtained "among the first ones" (how many have been opened? never mind)
• Two presents shown are the Shuffle and another, not the Winter Train, and will not be selected before getting the Winter Train

Assuming the Show Two was the first one selected (so at second attempt the # of Presents closed / Not Revealed drops from 17 to 15):

Attempt	# Presents Closed/Not Revealed	Probability of Win	Probability of Fail
1​	18​	5.56%​	94.44%​
2​	15​	6.67%​	93.33%​
3​	14​	7.14%​	92.86%​
4​	13​	7.69%​	92.31%​
5​	12​	8.33%​	91.67%​
6​	11​	9.09%​	90.91%​
7​	10​	10.00%​	90.00%​
8​	9​	11.11%​	88.89%​
9​	8​	12.50%​	87.50%​
10​	7​	14.29%​	85.71%​
11​	6​	16.67%​	83.33%​
12​	5​	20.00%​	80.00%​
13​	4​	25.00%​	75.00%​
14​	3​	33.33%​	66.67%​
15​	2​	50.00%​	50.00%​
16​	1​	100.00%​	0.00%​

Placing the selection of the Show Two at a second attempt or 3rd attempt will not change much the probabilities of attempts after it. Just the reduction from 17 to 15 will be seen as 16 to 14 at second attempt or from 15 to 13 at third attempt. At each attempt after that the Player will see one less present to open, odds to get the Winter Train will still grow slowly, but the increments will be better than at each previous turn. Big jumps of odds to get the Winter Train come when number of presents to open are 5 or less.

 

[Darrth Eugene Vader3]

100%
it will sure happen to some players
he already opened the see 2 (and a few other: "one among the first ones" so not really the 1st of all)
and is seeing 2
so let's say 10-15 unknown
1/10= 10%
1/15 = 6,67%
that are the odds of getting it as last AFTER already opening the show 2 in the first ones (1-5)
you can't check the last one (18th) if it is not the shuffle

100% & you can't check the last one (18th) if it is not the shuffle - To the poor luck Players at the last attempt 16th attempt (not the 18th due to the Show Two).
it will sure happen to some players - On the long run 5.56% will get it at first attempt, many others after more attempts.
so let's say 10-15 unknown - 15 unknown by 2nd to 3rd, 10 unknown by 7th attempt

 

[beelzebob666]

The probability of next present opened is the Winter Train is not 50%

that was not the question... the question was "what are the odds for the winter train kit to remain until the very last unopened present"

and that chance is 6.7% under the given circumstances (first present is show 2, showing the shuffle and not the DS) - aka 15 presents left

try	presents left	chance to find DS in try X	chance scaled to "not found before"	chance not found by try X	chance "found by try X or earlier"
1​	15​	6.7%​	6.7%​	93.3%​	6.7%​
2​	14​	7.1%​	6.7%​	86.7%​	13.3%​
3​	13​	7.7%​	6.7%​	80.0%​	20.0%​
4​	12​	8.3%​	6.7%​	73.3%​	26.7%​
5​	11​	9.1%​	6.7%​	66.7%​	33.3%​
6​	10​	10.0%​	6.7%​	60.0%​	40.0%​
7​	9​	11.1%​	6.7%​	53.3%​	46.7%​
8​	8​	12.5%​	6.7%​	46.7%​	53.3%​
9​	7​	14.3%​	6.7%​	40.0%​	60.0%​
10​	6​	16.7%​	6.7%​	33.3%​	66.7%​
11​	5​	20.0%​	6.7%​	26.7%​	73.3%​
12​	4​	25.0%​	6.7%​	20.0%​	80.0%​
13​	3​	33.3%​	6.7%​	13.3%​	86.7%​
14​	2​	50.0%​	6.7%​	6.7%​	93.3%​
15​	1​	100.0%​	6.7%​	0.0%​	100.0%​

if you are wondering about the chance scaled to "not found before" column...
try X only happens when on try X-1 the DS was not found - therefore the actual chance for that event has to be reduced

and in case you are wondering... on average, it is expected to find the DS after 5-6 presents

In general, the chances of that happening are 1/n for n remaining presents

 

[Darrth Eugene Vader3]

that was not the question...

I know that was not the original question, I was replying to a wrong answer given before.

50%
It happens, or it doesn't happen.

By DS you mean Winter Train, right?

Edit: Oh, DS = Daily special, ok.

 

[beelzebob666]

By DS you mean Winter Train, right?

Daily special, yes