Вивчайте Independence of Events

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Understanding independence is essential in probability theory. Two events, $A$ and $B$ , are said to be independent if the occurrence of one does not affect the probability of the other. Mathematically, events $A$ and $B$ are independent if and only if:

P(A ∩ B) = P(A) × P(B)

Here, $P(A ∩ B)$ is the probability that both $A$ and $B$ occur, and $P(A)$ and $P(B)$ are the probabilities of $A$ and $B$ occurring individually. If this equation does not hold, the events are dependent.

For instance, consider flipping two coins:

Let $A$ be the event "the first coin shows heads";
Let $B$ be the event "the second coin shows heads";
The probability of both coins showing heads is $P(A ∩ B) = 0.25$ ;
The product $P(A) × P(B) = 0.5 × 0.5 = 0.25$ ;
Since these values are equal, the two events are independent.

However, if you draw two cards from a deck without replacement, the event that the first card is an ace and the event that the second card is an ace are dependent, since removing an ace from the deck affects the probability of drawing another ace.


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import math

# Example 1: Coin flips (Independent)
p_a1, p_b1 = 0.5, 0.5
p_both1 = 0.25
print("Coin flips independent:", math.isclose(p_a1 * p_b1, p_both1))

# Example 2: Card draws without replacement (Dependent)
# Note: The marginal probability of the second card being an Ace is still 4/52
p_a2, p_b2 = 4/52, 4/52 
p_both2 = (4/52) * (3/51)
print("Card draws independent:", math.isclose(p_a2 * p_b2, p_both2))

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Секція 1. Розділ 6

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