Basic Probability Rules
In the previous lesson, we learned what probability is and how to calculate it for simple events. In real situations, events are often combined or related.
To handle such situations correctly, probability follows a few important rules. These rules help us calculate probabilities accurately and avoid mistakes.
Rule 1: Probability Range
The probability of any event must always lie between 0 and 1.
- Probability = 0 → Event is impossible
- Probability = 1 → Event is certain
Any probability value outside this range is incorrect.
Example
If someone claims that the probability of an event is 1.4 or −0.2, the calculation is wrong because probability cannot exceed 1 or be negative.
Rule 2: Sum of Probabilities Equals 1
The sum of probabilities of all possible outcomes of an experiment is always 1.
Numerical Example
When tossing a fair coin:
- Probability of Heads = 1/2
- Probability of Tails = 1/2
Total probability = 1/2 + 1/2 = 1
Rule 3: Complement Rule
The complement of an event is the event not happening.
The complement rule states:
P(Not A) = 1 − P(A)
Numerical Example
If the probability of rain today is 0.3, then:
Probability of no rain = 1 − 0.3 = 0.7
This rule is very useful when calculating probabilities of events that are easier to describe in terms of what does not happen.
Rule 4: Addition Rule (Mutually Exclusive Events)
Two events are mutually exclusive if they cannot happen at the same time.
For mutually exclusive events:
P(A or B) = P(A) + P(B)
Numerical Example
When rolling a die:
- Probability of getting 1 = 1/6
- Probability of getting 6 = 1/6
Getting 1 and getting 6 cannot happen together.
Probability of getting 1 or 6:
1/6 + 1/6 = 2/6 = 1/3
Rule 5: Addition Rule (Non-Mutually Exclusive Events)
If two events can happen at the same time, we must subtract the overlap to avoid double counting.
The rule becomes:
P(A or B) = P(A) + P(B) − P(A and B)
Real-World Example
In a group of students:
- Probability a student likes Math = 0.6
- Probability a student likes Science = 0.5
- Probability a student likes both = 0.3
Probability a student likes Math or Science:
0.6 + 0.5 − 0.3 = 0.8
Rule 6: Multiplication Rule (Independent Events)
Two events are independent if the occurrence of one does not affect the occurrence of the other.
For independent events:
P(A and B) = P(A) × P(B)
Numerical Example
Tossing a coin and rolling a die are independent events.
- Probability of Heads = 1/2
- Probability of rolling a 4 = 1/6
Probability of getting Heads and 4:
1/2 × 1/6 = 1/12
Summary of Probability Rules
| Rule | Description |
|---|---|
| Range Rule | Probability lies between 0 and 1 |
| Total Probability | Sum of all outcomes equals 1 |
| Complement Rule | P(Not A) = 1 − P(A) |
| Addition Rule | Used for “A or B” events |
| Multiplication Rule | Used for “A and B” events |
Practice Quiz
Question 1:
If P(A) = 0.65, what is P(Not A)?
Question 2:
Which rule is used for independent events?
Question 3:
Why do we subtract P(A and B) in the addition rule?
Mini Practice
A card is drawn from a standard deck of 52 cards.
- What is the probability of drawing a red card?
- What is the probability of not drawing a red card?
What’s Next
In the next lesson, we will study Independent and Dependent Events, which explains how events influence each other.