Draw one specific card!

[Must085]

Hi everyone,

New to this forum so I apologise in advance if I haven’t posted this in the right section.

I have to calculate the probability that a specific card (a Joker) will be drawn on blackjack when the total number of cards is 208 + 1 Joker = 209
On blackjack there can be from one to 6 boxes (customers) opened so there can be either 3, 5, 7, 9, 11 or 13 cards drawn per round.
What will be the probability of the joker being drawn for each of the cases where 3 to 13 cards are drawn?
Also, what will the frequency be of this joker being drawn i.e. every 30 rounds…

Hope it makes sense!

Thank you in advance!

 

[khansaheb]

Hi everyone,

New to this forum so I apologise in advance if I haven’t posted this in the right section.

I have to calculate the probability that a specific card (a Joker) will be drawn on blackjack when the total number of cards is 208 + 1 Joker = 209
On blackjack there can be from one to 6 boxes (customers) opened so there can be either 3, 5, 7, 9, 11 or 13 cards drawn per round.
What will be the probability of the joker being drawn for each of the cases where 3 to 13 cards are drawn?
Also, what will the frequency be of this joker being drawn i.e. every 30 rounds…

Hope it makes sense!

Thank you in advance!

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

 

[Steven G]

Why can't 4 cards be drawn? One player and the dealer--the player stands with 18 (say two 9's) and the dealer gets 2 picture cards.
Why is 13 the max? 6 players plus the dealer--why can't each player get 5 cards each? Suppose player 1, 2, 3 and 4 gets an A, 3, 5, 7 and 10, and player 5 and 6 gets 2, 4, 4, 6 and 10? This already accounts for 30 cards and doesn't even include the dealer's cards!

For some reason you think that each player gets exactly two cards and the dealer gets exactly one cards. In blackjack you don't have to stand with with your first two cards- and certainly the dealer can draw a 2nd card.

If you just want to assume that all players get two cards and the dealer gets one card that is perfectly fine. The problem is that you need to state this, as it goes against the way that blackjack is usually played.

 

[The Highlander]

Hi everyone,

New to this forum so I apologise in advance if I haven’t posted this in the right section.

I have to calculate the probability that a specific card (a Joker) will be drawn on blackjack when the total number of cards is 208 + 1 Joker = 209
On blackjack there can be from one to 6 boxes (customers) opened so there can be either 3, 5, 7, 9, 11 or 13 cards drawn per round.
What will be the probability of the joker being drawn for each of the cases where 3 to 13 cards are drawn?
Also, what will the frequency be of this joker being drawn i.e. every 30 rounds…

Hope it makes sense!

Thank you in advance!

If you just want to assume that all players get two cards and the dealer gets one card that is perfectly fine. The problem is that you need to state this, as it goes against the way that blackjack is usually played.

It's my understanding too that, in Blackjack, every player (including the dealer) gets two cards, so there should be either 4, 6, 8, 10, 12 or 14 cards drawn per round.

Can you explain this apparent anomaly?

Also, I suspect you meant "e.g. every 30 rounds" not "i.e. every 30 rounds" when you asked about "frequency" or did you specifically want to know the probability of one or more (if there was more than 2 players) joker(s) being drawn in exactly 30 rounds which would clearly change unless there was a fixed number of players throughout those 30 rounds.

And, of course, the whole calculation (except if you consider only the first round from a newly stacked deck) is further complicated (and to my mind rendered impossible ) by the fact that, although 2 to 14 cards (or 3 to 13 in your description) may be dealt initially, the players will almost certainly demand an indeterminable number of extra cards to be drawn in each round! (At least one of them is bound to say: "Hit me.")

 

[Must085]

Hi all, thanks for your replies and questions
raised to try and understand what I’m trying to achieve.
I apologise if I haven’t been clear and my explanation was confusing.
In a nutshell what I would need to calculate is an estimated frequency for this joker to appear every day throughout an average of 500 hands of blackjack dealt per day.
This is the main objective.
The reason for that is because every time a joker appear the player who received the joker wins a bonus prize.
If the dealers draw a joker for themselves than there is no bonus to be paid.

Now, I am aware that players can pull cards and might not stand on their first two cards received.
My previous calculation was assuming the cards drawn on the initial deal only and therefore for one box played = 3 cards, two for the player and one for the dealer and so on up until a total of 6 boxes in play = 13 cards in total.
What I would like to know is just the estimated frequency the joker will appear if one box is played up until six boxes played but I assume this frequency would also include when it obviously appears as the dealer card and therefore that probability should be discarded because of no bonus prize being unlocked and therefore I can disregard that.

From the original post I then made another assumption where instead of taking jnto consideration just the initial deal (just two cards per player) I added one more card per box (in case the player will decide do hit and not stand) and eliminated the dealer’s first card as this even if it appears there would not count.
This assumption then led me to the below numbers and the corresponding estimated probabilities:

1 box = 3 cards = 1.4%
2 = 6 cards = 2.8%
3 = 9 cards = 4.3%
4 = 12 = 5.7%
5 = 15 = 7.1%
6 = 18 = 8.6%

Would that be more or less correct?
Would that mean that in let’s say 1000 hands the joker would appear 14 times if 1 box and therefore 1 player was playing?
Thanks in advance for your help!

 

[Steven G]

It's my understanding too that, in Blackjack, every player (including the dealer) gets two cards, so there should be either 4, 6, 8, 10, 12 or 14 cards drawn per round.

Can you explain this apparent anomaly?

Actually the dealer gets one (up) card. After all the players are done, then the dealer takes their 2nd card (unless everyone already lost)

 

[Agent Smith]

For [imath]3[/imath] draws, the probability that none are jokers = [imath]\frac{207}{208} \times \frac{206}{207} \times \frac{205}{206}[/imath]

 

[The Highlander]

Actually the dealer gets one (up) card. After all the players are done, then the dealer takes their 2nd card (unless everyone already lost)

No, no, no!
There is always an even number of cards drawn at the start of a game of Blackjack.

(I have set these videos up to begin at the relevant place.)

Watch 20 seconds of this video...

and/or

watch 10 seconds of this video...

Every player and the dealer gets two cards at the start; although you are quite right that one of the dealer's cards is turned face up but s/he still has two cards on the table.

The OP got an answer elsewhere that, no doubt, satisfied him/her despite being wrong in more ways than one, so we're not likely to see or hear from him/her again.

I have, therefore, posted the correct probabilities below.

However, the point I was trying to make (above) but which was, ostensibly, ignored, is that the probability of a Joker turning up can only be determined for the first round of play!

In the first round of play anything between 4 and 10 cards may be drawn from the deck if there is only one box being played and anything between 14 and 35 (or possibly even more) cards might be drawn if all six boxes are in play!

Therefore, since, even after a single round of play, there is an indeterminate number of cards left in the deck, any calculation of the probability of a Joker being drawn in subsequent rounds of play or, indeed, the frequency of its appearance over any number of rounds, is simply impossible!

Here are the probabilities of the Joker being drawn on the first round from a well shuffled deck of 209 cards comprising four packs of (52) cards plus 1 Joker...

Number of boxes being played​	Probability (Rounded to 2 d.p.)​
1​	1.91%​
2​	2.87%​
3​	3.83%​
4​	4.79%​
5​	5.74%​
6​	6.70%​

 

[The Highlander]

For [imath]3[/imath] draws, the probability that none are jokers = [imath]\frac{207}{208} \times \frac{206}{207} \times \frac{205}{206}[/imath]

(Once again!) I am unsure of the relevance of your post!

Do you just skim through a thread and, when you notice something you reckon you know a bit about, post whatever comes into your mind?

 

[Agent Smith]

(Once again!) I am unsure of the relevance of your post!

Do you just skim through a thread and, when you notice something you reckon you know a bit about, post whatever comes into your mind?

Sumthin like that, si!

 

[Must085]

No, no, no!
There is always an even number of cards drawn at the start of a game of Blackjack.

(I have set these videos up to begin at the relevant place.)

Watch 20 seconds of this video...

and/or

watch 10 seconds of this video...

Every player and the dealer gets two cards at the start; although you are quite right that one of the dealer's cards is turned face up but s/he still has two cards on the table.

The OP got an answer elsewhere that, no doubt, satisfied him/her despite being wrong in more ways than one, so we're not likely to see or hear from him/her again.

I have, therefore, posted the correct probabilities below.

However, the point I was trying to make (above) but which was, ostensibly, ignored, is that the probability of a Joker turning up can only be determined for the first round of play!

In the first round of play anything between 4 and 10 cards may be drawn from the deck if there is only one box being played and anything between 14 and 35 (or possibly even more) cards might be drawn if all six boxes are in play!

Therefore, since, even after a single round of play, there is an indeterminate number of cards left in the deck, any calculation of the probability of a Joker being drawn in subsequent rounds of play or, indeed, the frequency of its appearance over any number of rounds, is simply impossible!

Here are the probabilities of the Joker being drawn on the first round from a well shuffled deck of 209 cards comprising four packs of (52) cards plus 1 Joker...

Number of boxes being played​	Probability (Rounded to 2 d.p.)​
1​	1.91%​
2​	2.87%​
3​	3.83%​
4​	4.79%​
5​	5.74%​
6​	6.70%​

What you are describing is a variant of Blackjack, to be precise the variant that is played in the majority of the US. On the initial deal on blackjack there are not ALWAYS an even number of cards, in the majority of countries in Europe the dealer does not draw a second card face down for themselves but draws a second card only after all the players have made a decision on whether to hit or stand.

 

[Steven G]

No, no, no!
There is always an even number of cards drawn at the start of a game of Blackjack.

(I have set these videos up to begin at the relevant place.)

Watch 20 seconds of this video...

and/or

watch 10 seconds of this video...

Every player and the dealer gets two cards at the start; although you are quite right that one of the dealer's cards is turned face up but s/he still has two cards on the table.

The OP got an answer elsewhere that, no doubt, satisfied him/her despite being wrong in more ways than one, so we're not likely to see or hear from him/her again.

I have, therefore, posted the correct probabilities below.

However, the point I was trying to make (above) but which was, ostensibly, ignored, is that the probability of a Joker turning up can only be determined for the first round of play!

In the first round of play anything between 4 and 10 cards may be drawn from the deck if there is only one box being played and anything between 14 and 35 (or possibly even more) cards might be drawn if all six boxes are in play!

Therefore, since, even after a single round of play, there is an indeterminate number of cards left in the deck, any calculation of the probability of a Joker being drawn in subsequent rounds of play or, indeed, the frequency of its appearance over any number of rounds, is simply impossible!

Here are the probabilities of the Joker being drawn on the first round from a well shuffled deck of 209 cards comprising four packs of (52) cards plus 1 Joker...

Number of boxes being played​	Probability (Rounded to 2 d.p.)​
1​	1.91%​
2​	2.87%​
3​	3.83%​
4​	4.79%​
5​	5.74%​
6​	6.70%​

You seem to be correct about the dealer getting two cards these days. I might be wrong but I recall that in the 80's that the dealer only got one card. Thanks for updating this for me!