Expected Value: Cards

Calculate expected value for card scenarios

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Expected Value in Blackjack

A♥

K♠

= 21!

Blackjack Rules

Number cards: face value (2-10)
Face cards (J, Q, K): 10
Ace: 1 or 11 (player choice)
Goal: Get as close to 21 without going over

Expected Value of Initial Hand

You receive 2 cards. What is the expected hand value?

E(one card) with Ace = 11

Using Ace = 11 (optimal for starting hands):

E(two cards) by Linearity

Note: This is approximate since cards are drawn without replacement.

Initial Hand Expected Value

E(Hand) ≈ 14.6

On average, your starting hand is worth about 14-15 points.

Probability of Natural Blackjack

Blackjack (21 with 2 cards) requires Ace + 10-value card.

Aces:

4 cards

10-value cards:

16 cards (10, J, Q, K × 4 suits)

Calculation

≈ 4.83% or about 1 in 21 hands

Expected Value for the Player

In standard blackjack with basic strategy:

The house edge is about 0.5% with perfect basic strategy. This makes blackjack one of the fairest casino games.

Key Insight

Blackjack has a low house edge because players can make decisions (hit, stand, double, split) that affect outcomes. Using basic strategy optimizes expected value and nearly levels the playing field.