What is the difference between the product rule for independent and dependent events?

Independent Events The probability of one event happening isn't influenced by the outcome of another.
In mathematical terms
A
,
B
events
,
P
(
A
∩
B
)
=
P
(
A
)
⋅
P
(
B
)

This reads: Probability of A and B happening is equal to the probability of A happening multiplied by the probability of B happening
An equivalent definition
P
(
A
∣
B
)
=
P
(
A
)
This reads: The probability of A given B happened is equal the probability of A
Example:
Event A: Rolling a number larger than four on a die in the first roll Event B: Rolling a 6 on a die on the second roll. These events are independent because one roll of a dice doesn't influence the outcome of another dice roll.
Dependent Events Is the oposite.If the outcome of one event influences the probability of the other they aren't independent
P
(
A
∣
B
)
≠
P
(
A
)
Example:
Event A: Rolling a number larger than four on a die Event B: Rolling a 6 on a die on the same roll. These events are dependent because ,as we are in the same dice roll, the occurrence of event B changes the probability of event A, actually it makes it 1.

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