To fully grasp the
multiplication rule of probabilities, it helps to divide the concept into two
key rules:
Rule #1: Multiplication of Independent Events
Rule #2: Multiplication of
Dependent Events
Independent
Events
Two events, A and B,
are independent if the occurrence of one (A or B) does not
affect the occurrence of the other (B or A).
Example
When rolling two dice, the
result of one die has no influence on the result of the other. We say these
events are independent.
Rolling
two dice: The outcome of one does not affect the other
In real life, many events
are independent. For example:
·
"Raining today"
and "Tomorrow being a holiday" are independent because rain today
doesn’t change the likelihood of tomorrow being a holiday, nor does a holiday
tomorrow affect the chance of rain today.
· In healthcare:
o A person
being nearsighted doesn’t affect their probability of having cavities.
o A
person’s profession doesn’t influence their likelihood of developing kidney
stones.
o Marital
status doesn’t change the probability of being bald.
Multiplication
Rule #1 (for Independent Events)
If A and B are independent,
the probability of both A and B occurring is the product of
their individual probabilities:
P (A ⋃ B) = P(A)×P(B)
Example
You roll two dice
simultaneously—one red and one yellow. What is the probability of getting
a 3 on the yellow die and a 5 on the red die?
Using Rule #1:
If the occurrence of event
A changes the probability of event B occurring, the two events
are dependent.
Example
A drawer contains six
socks: three red and three blue. You want a pair of red socks. Without looking,
you pull out one sock—it’s red. Without replacing it, you draw a second sock.
Now, the probability of the second sock being red is lower. Why?
Removing a red sock changes the probability for the next draw
The probability changed
because the first event affected the second. Thus, these are dependent
events.
In real life, dependent
events are common:
·
Smoking increases the
probability of lung cancer.
·
Drunk driving raises the
chance of traffic accidents.
·
Vaccination reduces the
probability of contracting a disease.
Conditional
Probability
The conditional
probability of B given A (P(B∣A) is the
probability of event B occurring after A has already occurred.
Example
Using the sock example:
The probability of both socks being red is:
Another
Example
A fair die is rolled.
1. What is
the probability of rolling a 5?
2. What is the probability of rolling a 5, given an odd number was rolled?
The sample space is now reduced to
{1, 3, 5}
If A and B are dependent,
the probability of both occurring is:
P (A ∩ B) =P(A)×P
(B∣A
Example
A hat contains five
balls—three blue and two red. Two balls are drawn randomly without
replacement. What is the probability that both are red?
·
Condition for Independence
Two events are
independent if and only if the probability of both occurring
equals the product of their individual probabilities. In other words, the
occurrence of one provides no information about the other.
Example
A coin is flipped twice:
·
The outcome of the first
flip does not influence the second.
·
Thus, the two events are
independent.
Real world exemples:
"Tomorrow is a holiday" doesn’t
alter "Raining today": independent events.
“Smoking”
increases “lung cancer risk": dependent events
“Myopia”
doesn’t alter “cavity risk”: independent events.
"Obesity increases the probability of diabetes: dependent events.
"Having migraines” doesn’t change your probability of “developing arthritis": independent events
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