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Two events are independent if the outcome of one doesn't impact the other.
One event occurring does not impact the probability of another event occurring.
Scenario: Considering the following two events, determine if they're independent.
- Event A: Roll a number greater than 4
- Event B: Roll an odd number
If any are true, A and B are independent.
P(B | A) (B given A) = P(B)
P(A | B) (A given B) = P(A)
P(A ∩ B) = P(A) x P(B)
SA = {1, 2, 3, 4, 5, 6}
Two out of the six potential outcomes fall in Event A...
P(A) = 2 / 6
P(A) = 1/3
SB = {1, 2, 3, 4, 5, 6}
Three out of the six potential outcomes fall in Event B...
P(B) = 3 / 6
P(B) = 1/2
P(B | A) = P(B)
It represents the probability of B occurring, given A has occurred.
SB | A = {1, 2, 3, 4, 5, 6}
P(B | A) = 1 / 2
P(B | A) = 1/2
Answer: Rolling an odd number given that you've rolled a number greater than 4 is 1/2... the same probability as rolling an odd number outright (1/2). In other words... these events are independent!
P(A | B) = P(A)
It represents the probability of A occurring, given B has occurred.
SA | B = {1, 2, 3, 4, 5, 6}
P(A | B) = 1 / 3
P(A | B) = 1/3
Answer: Rolling a number greater than 4 given that you've rolled an odd number is 1/3... the same probability as rolling a number greater than 4 outright (1/3). In other words... these events are independent!
P(A ∩ B) = P(A) x P(B)
SA ∩ B = {1, 2, 3, 4, 5, 6}
P(A ∩ B) = 1 / 6
P(A ∩ B) = 1/6
P(A ∩ B) = P(A) x P(B)
1/6 = 1/3 x 1/2
1/6 = 1/6
Yes, they are independent!
PRO TIP: This is one of those cases that using fractions works better so it's exactly the same. That's why I didn't use decimals.
Give me a scenario that's not independent...
Scenario: Considering the following two events, determine if they're independent.
- Event A: Roll a number greater than 3
- Event B: Roll an odd number
P(B | A) = P(B)
P(A | B) = P(A)
P(A ∩ B) = P(A) x P(B)
SA = {1, 2, 3, 4, 5, 6}
Three out of the six potential outcomes fall in Event A...
P(A) = 3 / 6
P(A) = 1/2
SA | B = {1, 2, 3, 4, 5, 6}
One out of the three Event B outcomes fall in Event A...
P(A | B) = 1 / 3
P(A | B) = 1/3
Answer: Rolling a number greater than 3 given that you've rolled an odd number is 1/3... a different probability than rolling a number greater than 3 outright (1/2). In other words... these events are not independent!
