Lära Independence of Events

Avsnitt 1. Kapitel 6

single

Svep för att visa menyn

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.


              123456789101112
            
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))

Byt till skrivbordet för praktisk övningFortsätt där du är med ett av alternativen nedan

Var allt tydligt?

Tack för dina kommentarer!

Avsnitt 1. Kapitel 6

single

Fråga AI

Fråga vad du vill eller prova någon av de föreslagna frågorna för att starta vårt samtal