Are the Events a Face Up Card Is a Black Jack and B Get a True Blackjack Independent?

When playing the popular casino game of blackjack, there are certain events that can occur that can greatly affect the outcome of the game. Two such events are when a player is dealt a face up card that is an ace or a ten-point card (a “blackjack”), and when a player gets a total hand value of exactly 21 (a “true blackjack”).

The question is, are these two events independent of each other In other words, does the occurrence of one event affect the probability of the other event occurring

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Probability of Getting a Face Up Blackjack

Let’s first look at the probability of getting a face up blackjack. To do this, we need to consider all possible combinations of cards that can be dealt to a player.

  • If the first card dealt to the player is an ace, there are 16 ten-point cards remaining in the deck out of 51 cards left. This means there is a 16/51 chance (approximately 31%) that the next card will be worth ten points and give the player a blackjack.
  • If the first card dealt to the player is worth ten points, there are four aces remaining in the deck out of 51 cards left. This means there is a 4/51 chance (approximately 8%) that the next card will be an ace and give the player a blackjack.

Therefore, combining these two probabilities gives us:

P(face up blackjack) = P(ace as first card) x P(ten-point card as second card) + P(ten-point card as first card) x P(ace as second card)

P(face up blackjack) = (4/52) x (16/51) + (16/52) x (4/51)

P(face up blackjack) = 64/2652 + 64/2652

P(face up blackjack) = 128/2652

P(face up blackjack) = approximately 4.8%

Probability of Getting a True Blackjack

Next, let’s look at the probability of getting a true blackjack. This occurs when a player is dealt an ace and a ten-point card as their first two cards.

There are several ways to calculate the probability of this event, but one common method is to consider all possible two-card combinations that can be dealt from a single deck of cards.

  • There are four aces and 16 ten-point cards in a deck, so the number of ways to choose one ace and one ten-point card is:
    • C(4,1) x C(16,1) = 4 x 16 = 64
  • The total number of possible two-card combinations that can be dealt is:
    • C(52,2) = (52 x 51)/2 = 1326

Therefore, the probability of getting a true blackjack is:

P(true blackjack) = Number of ways to get an ace and a ten-point card / Total number of possible two-card combinations

PRO TIP:When it comes to a Face Up card being a Black Jack and B getting a True Blackjack, they are independent events. This means that neither event has an effect on the outcome of the other. The probability of each event remains the same regardless of whether the other one has occurred or not.

P(true blackjack) = 64 / 1326

P(true blackjack) = approximately 4.8%

Independence of Events

To determine whether the events of getting a face up blackjack and getting a true blackjack are independent, we need to compare the probabilities of each event occurring separately to the probability of both events occurring together.

If the two events are independent, then the probability of both events occurring together should be equal to the product of the probabilities of each event occurring separately.

Let’s calculate this:

P(face up blackjack) x P(true blackjack) = (128/2652) x (64/1326)

P(face up blackjack) x P(true blackjack) = 8192/7030152

P(face up blackjack and true blackjack) = Number of ways to get a face up blackjack and a true blackjack / Total number of possible two-card combinations

There is only one way to get a face up blackjack and a true blackjack, which is to be dealt an ace and a ten-point card as your first two cards when the dealer’s face up card is also worth ten points. Therefore:

P(face up blackjack and true blackjack) = 1 / 1326

Comparing these two probabilities, we can see that:

P(face up blackjack) x P(true blackjack) ≠ P(face up blackjack and true blackjack)

This means that the events of getting a face up blackjack and getting a true blackjack are dependent on each other. The occurrence of one event affects the probability of the other event occurring.

Conclusion

In conclusion, while it may seem that getting a face up card with value ten or an ace is independent from getting a true black jack, our analysis shows that they are actually dependent on each other. The occurrence of one event does affect the probability of the other event occurring. This information can be valuable for players who want to improve their blackjack strategy and maximize their chances of winning.

10 Related Question Answers Found

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Can You Split Different Face Cards Blackjack?

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Can I Split Face Cards in Blackjack?

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