Understanding Dependent Random Variables
Introduction
In many situations, outcomes are random, but sometimes they are connected. Random quantities that are linked are called dependent random variables. Understanding these dependencies helps in making better decisions, from finance to games like poker.
Dependent Random Variables
Two random variables are dependent if knowing the value of one gives information about the other.
For example:
- \(X\) = the cards you are dealt in poker
- \(Y\) = the probability of winning the hand
In this case, knowing \(X\) affects \(Y\), so they are dependent. Mathematically:
$$P(Y \text{ happens } | X \text{ happened}) \neq P(Y \text{ happens})$$
Poker Example: Texas Hold’em
Suppose we are playing Texas Hold’em with a full table of 9 players. Let’s calculate the probability of winning given a strong starting hand.
Step 1: Define the variables
- \(X\) = being dealt a pair of aces (AA) as hole cards
- \(Y\) = winning the hand
Step 2: Probability of getting AA
In a 52-card deck, there are 4 aces. The number of ways to get 2 aces:
$${4 \choose 2} = 6$$
The total number of 2-card combinations:
$${52 \choose 2} = 1326$$
So, the probability of being dealt AA:
$$P(X) = \frac{6}{1326} \approx 0.0045 \text{ (0.45% of hands)}$$
Step 3: Probability of winning given AA
Statistical simulations and poker analysis suggest that AA is the strongest starting hand, with about a 31% chance to win at a 9-player table:
$$P(Y | X) \approx 0.31$$
Step 4: Probability of winning without any information (random hand)
The average probability of winning a hand with a random hand of 2 cards at a 9-player table is approximately:
$$P(Y) \approx \frac{1}{9} \approx 0.111$$
Step 5: Dependency check
Since:
$$P(Y | X) = 0.31 \neq 0.111 = P(Y)$$
The probability of winning changes significantly when we know we have AA. This confirms that X (hole cards) and Y (winning) are dependent random variables.
Step 6: Expected Value Example
Suppose a player bets $10 before the flop. The expected return depends on the probability of winning:
$$E[\text{win}] = P(Y | X) \times 10 = 0.31 \times 10 = 3.1$$
For a random hand:
$$E[\text{win}] = P(Y) \times 10 = 0.111 \times 10 = 1.11$$
This shows how knowing your hand changes your expected outcome and why the variables are dependent.
Other Examples of Dependent Variables
- Stock Prices: Two tech company stocks may rise or fall together because they are affected by the same market trends.
- Weather and Umbrellas: As mentioned, rainfall affects umbrella use.
- Health Measurements: Two related health indicators, like blood pressure and cholesterol, may move together because of lifestyle factors.
Table: Winning Probability by Starting Hand (9-Player Table)
| Starting Hand (X) | Probability of Winning (Y) | Notes |
|---|---|---|
| AA | 31% | Strongest possible hand |
| KK | 23% | Vulnerable to AA |
| 19% | Strong but can lose to overpairs | |
| JJ | 13% | Vulnerable to higher pairs and overcards |
| TT | 11% | Medium strength, often needs improvement |
| 99 | 9% | Can hit sets to win, but generally weaker |
| 88 | 8% | Relies on flopping a set |
| AK suited | 17% | High potential, strong top pairs and straights |
| AQ suited | 14% | Strong but can be dominated by AK or higher pairs |
| AJ suited | 12% | Moderate strength, can make top pair or straight |
| KQ suited | 11% | Good for straight or flush possibilities |
| QJ suited | 10% | Decent for straights, less likely to win high pairs |
| AK offsuit | 16% | High potential but slightly weaker than suited |
| AQ offsuit | 13% | Similar to suited but slightly less chance |
| Random Hand | 11% | Average probability of winning with any hand |
Interpretation:
- Top Pairs: AA, KK, and QQ dominate, with probabilities dropping quickly for lower pairs.
- Suited Connectors: AK, AQ, and KQ suited have higher potential than their offsuit counterparts due to flush possibilities.
- Smaller Pairs: 88–99 can win if they hit sets but are generally weaker, showing lower probability.
- Random Hands: On average, a random hand has about an 11% chance to win in a 9-player game, highlighting how much knowing your cards affects your winning probability.
This table gives a view of dependency: the probability of winning (Y) depends strongly on the starting hand (X). The stronger the hand, the higher the chance of winning.
Conclusion
Dependent random variables are everywhere, from poker to finance. In poker, knowing your hole cards drastically changes your probability of winning. Understanding these relationships helps in predicting outcomes and making better decisions.
