Basic statistics question

Why is 10 the most likely card to draw?

I realize that there are more 10s than any other single card. So if the question is 10 or 5, 10 or 7, etc. the odds are always in favor of the 10. I also realize that for any given deal you can only draw a 10, 5, 7, etc. You can't draw not-10, because not-10 isn't a card.

On the other hand, there are more not-10s than 10s.

So, it seems that it is true both that you are more likely to draw a 10 than any other particular card, and that you are more likely to draw a not-10 than a 10. How can we reconcile these two? And why does the consensus of the whole world rest with the former?

A 2 3 4 5 6 8 9 10 J Q K.

You are not most likely to draw a 10. There is a 4/13 chance of getting a ten, which is significantly higher than all the others at 1/13.

Answer: semantics.

A 10 is "the most likely card to draw" because if you compare each number separately, you will see that the specific number that you are most likely to draw is a 10.

That doesn't mean that it is necessarily very likely to be drawn - just that it is more likely than any other specified number.

You can do the same thing with two dice - a 7 is more likely than a 2, 3, 4, 5, 6, 8, 9, 10, 11, or 12, even though you only have a 1/6 chance of rolling a 7.

Nice! So it's a false dilemma, generated by "intuition." It seems like if 10 is more likely than any other number, it should also be the most likely to draw, but not so. Excellent. I feel now a little closer to understanding the nature of reality (but not much).

That's the same idea as throwing a 7 with two dice. It's the most likely outcome but it only occurs 1/6 of the time. "Most likely" does not imply "majority."

Quote: ueatuafish

Nice! So it's a false dilemma, generated by "intuition." It seems like if 10 is more likely than any other number, it should also be the most likely to draw, but not so.

There is a simplified blackjack strategy that the Wizard analyzed under his "Bad Strategies".

Since a 10 is the most likely card to be the dealer's hold card, some people simply "Assume a ten in the hole":

For this strategy the Wizard first figured out the optimal basic strategy under this assumption. If the dealer had an ace up, then I reverted to proper basic strategy, because the dealer would have peeked for blackjack, making a 10 impossible. This "assume a ten" strategy results in a house edge of 10.03%.

Rather than just using "assume a 10" the Wizard developed a simple strategy where the cost due to imperfect plays is 0.14% only. It's for people who can't be bothered to learn complete proper strategy, but don't want to make a lot of effort.

Quote: pacomartin

Since a 10 is the most likely card to be the dealer's hold card, some people simply "Assume a ten in the hole":

For this strategy the Wizard first figured out the optimal basic strategy under this assumption. If the dealer had an ace up, then I reverted to proper basic strategy, because the dealer would have peeked for blackjack, making a 10 impossible. This "assume a ten" strategy results in a house edge of 10.03%.

But I pray to God that anyone that follows this strategy stands still on all hard 17s or better, which the Wizard's sim didn't always do. Also after about 5 minutes at the blackjack table, they will be chastised enough by the table into standing in that spot.