Help with some maths

How do I calculate the probability of something happening, when two events occur.
For example how do I calculate the probability of getting 21 in blackjack when doubling on 11.
I understand I have to first calculate the probability of getting 11 in my first two cards, and then I have to calculate the probability of getting a ten for my 3rd card. How do I combine these two probabilities to calculate the overall probability.

Cheers

Quote: kingcreights

How do I calculate the probability of something happening, when two events occur.
For example how do I calculate the probability of getting 21 in blackjack when doubling on 11.
I understand I have to first calculate the probability of getting 11 in my first two cards, and then I have to calculate the probability of getting a ten for my 3rd card. How do I combine these two probabilities to calculate the overall probability.

Cheers

This link will help explain how to combine the two probabilities.

Here's how I would do your example for a 6-deck shoe (312 cards) assuming you will double down 11 vs any dealer's up card. The probability of getting a 2-card 11 off the top of the shoe is this sum: P(11) = P(2)*P(9|2) + P(3)*P(8|3) + P(4)*P(7|4) + P(5)*P(6|5) + P(6)*P(5|6) + P(7)*P(4|7) + P(8)*P(3|8) + P(9)*P(2|9) = 8 * (24/312) * (24/311) = 0.04749. So the probability of getting 11, doubling down, and getting 21 is P = P(10|11)*P(11) = (96/310) * P(11) = 0.01471.

There are 8 cards (2-9) that can be your first card to allow for a two-card eleven, and only 1 card for the second card. To find the probability of getting the first card being a 2-9 is (8*6*4)/312, since there are 8 cards, 6 decks, and 4 suits. For the second card, there is only one possible card to make a two-card eleven (if first is 2, then second card can only be 9. If first is 3 second card can only be 8, etc.). So it'd be (1*6*4)/311. And for the third card to be a ten (TJQK), you'd do (4*6*4)/310.

Note the 312, 311, and 310 are the number of cards to pick from (6 decks * 52 cards).

(8*6*4)/312 * (1*6*4)/311 * (4*6*4)/310
=
(8*6*4*6*4*4*6*4) / (312*311*310)
=
0.01470642209
~ 1.47% chance of it occurring.

Quote: kingcreights

how do I calculate the probability of getting 21 in blackjack when doubling on 11

The answer to this question is much simpler that the previous answers.

How many "10" cards are left in the deck? 4*4*6 = 96
How many cards left are left in the deck? (52*6) - 2 = 310

Probability = 96/310 = 30.97%.

The previous two answers have actually answered a different question - What is the chance of getting 11 on my first two cards AND then drawing to 21?

Do you suppose reading the OP would be of help?

Quote: DeMango

Do you suppose reading the OP would be of help?

Who is this directed at?

Quote: Peregrine

Who is this directed at?

You. The question at hand WAS what is the probability of getting 21 given that my first two cards equal 11:-)

Quote: Peregrine

The answer to this question is much simpler that the previous answers.

How many "10" cards are left in the deck? 4*4*6 = 96
How many cards left are left in the deck? (52*6) - 2 = 310

Probability = 96/310 = 30.97%.

The previous two answers have actually answered a different question - What is the chance of getting 11 on my first two cards AND then drawing to 21?

That's ironic.

According to Djatc it's 50%.... Either you get it or you don't.

Quote: AxelWolf

According to Djatc it's 50%.... Either you get it or you don't.

Quote: djatc

He meant to post something, really he did. But he was busy...;-)

Quote: TwoFeathersATL

He meant to post something, really he did. But he was busy...;-)

It was a 50/50 shot. Either he was going to say something, or he wasn't.

Quote: beachbumbabs

It was a 50/50 shot. Either he was going to say something, or he wasn't.

Sly answer darling. You know I have a teenage crush on you right?
Just 2F ;-)