In this section, we will delve into the fundamental rules of probability that are essential for understanding and solving problems in statistics. These rules form the backbone of probability theory and are crucial for making inferences about data.
Key Concepts
- Sample Space (S): The set of all possible outcomes of a random experiment.
- Event (E): A subset of the sample space. An event can consist of one or more outcomes.
- Probability of an Event (P(E)): A measure of the likelihood that the event will occur, ranging from 0 to 1.
Basic Probability Rules
- The Addition Rule
The addition rule is used to find the probability of the union of two events. There are two forms of the addition rule:
a. For Mutually Exclusive Events
If two events, A and B, are mutually exclusive (they cannot occur at the same time), the probability of either event occurring is the sum of their individual probabilities.
\[ P(A \cup B) = P(A) + P(B) \]
Example: If the probability of drawing a red card from a deck of cards is \( \frac{1}{2} \) and the probability of drawing a black card is \( \frac{1}{2} \), then the probability of drawing either a red card or a black card is:
\[ P(\text{Red} \cup \text{Black}) = P(\text{Red}) + P(\text{Black}) = \frac{1}{2} + \frac{1}{2} = 1 \]
b. For Non-Mutually Exclusive Events
If two events, A and B, are not mutually exclusive (they can occur at the same time), the probability of either event occurring is the sum of their individual probabilities minus the probability of both events occurring.
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Example: If the probability of a student passing math is 0.7 and the probability of passing science is 0.6, and the probability of passing both is 0.5, then the probability of passing either math or science is:
\[ P(\text{Math} \cup \text{Science}) = P(\text{Math}) + P(\text{Science}) - P(\text{Math} \cap \text{Science}) = 0.7 + 0.6 - 0.5 = 0.8 \]
- The Multiplication Rule
The multiplication rule is used to find the probability of the intersection of two events. There are two forms of the multiplication rule:
a. For Independent Events
If two events, A and B, are independent (the occurrence of one does not affect the occurrence of the other), the probability of both events occurring is the product of their individual probabilities.
\[ P(A \cap B) = P(A) \times P(B) \]
Example: If the probability of flipping a coin and getting heads is \( \frac{1}{2} \) and the probability of rolling a die and getting a 6 is \( \frac{1}{6} \), then the probability of both getting heads and rolling a 6 is:
\[ P(\text{Heads} \cap \text{6}) = P(\text{Heads}) \times P(\text{6}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} \]
b. For Dependent Events
If two events, A and B, are dependent (the occurrence of one affects the occurrence of the other), the probability of both events occurring is the product of the probability of the first event and the conditional probability of the second event given that the first event has occurred.
\[ P(A \cap B) = P(A) \times P(B|A) \]
Example: If the probability of drawing an ace from a deck of cards is \( \frac{4}{52} \) and the probability of drawing a king after drawing an ace (without replacement) is \( \frac{4}{51} \), then the probability of drawing an ace and then a king is:
\[ P(\text{Ace} \cap \text{King}) = P(\text{Ace}) \times P(\text{King}|\text{Ace}) = \frac{4}{52} \times \frac{4}{51} = \frac{16}{2652} = \frac{4}{663} \]
- The Complement Rule
The complement rule is used to find the probability of the complement of an event. The complement of an event A is the event that A does not occur, denoted as \( A' \).
\[ P(A') = 1 - P(A) \]
Example: If the probability of it raining tomorrow is 0.3, then the probability of it not raining is:
\[ P(\text{No Rain}) = 1 - P(\text{Rain}) = 1 - 0.3 = 0.7 \]
Practical Exercises
Exercise 1: Addition Rule for Mutually Exclusive Events
Two dice are rolled. What is the probability of getting a sum of 2 or a sum of 12?
Solution:
- The probability of getting a sum of 2 (only one combination: (1,1)) is \( \frac{1}{36} \).
- The probability of getting a sum of 12 (only one combination: (6,6)) is \( \frac{1}{36} \).
Since these events are mutually exclusive:
\[ P(2 \cup 12) = P(2) + P(12) = \frac{1}{36} + \frac{1}{36} = \frac{2}{36} = \frac{1}{18} \]
Exercise 2: Multiplication Rule for Independent Events
A bag contains 5 red balls and 3 blue balls. Two balls are drawn with replacement. What is the probability of drawing two red balls?
Solution:
- The probability of drawing a red ball on the first draw is \( \frac{5}{8} \).
- Since the ball is replaced, the probability of drawing a red ball on the second draw is also \( \frac{5}{8} \).
Since these events are independent:
\[ P(\text{Red} \cap \text{Red}) = P(\text{Red}) \times P(\text{Red}) = \frac{5}{8} \times \frac{5}{8} = \frac{25}{64} \]
Exercise 3: Complement Rule
The probability of a student passing an exam is 0.85. What is the probability of the student failing the exam?
Solution:
\[ P(\text{Fail}) = 1 - P(\text{Pass}) = 1 - 0.85 = 0.15 \]
Common Mistakes and Tips
- Misinterpreting Mutually Exclusive Events: Remember that mutually exclusive events cannot happen at the same time. If there's any overlap, they are not mutually exclusive.
- Forgetting to Subtract the Intersection: When using the addition rule for non-mutually exclusive events, always subtract the intersection to avoid double-counting.
- Confusing Independent and Dependent Events: Ensure you understand whether events are independent or dependent before applying the multiplication rule.
Conclusion
Understanding and applying the basic rules of probability is essential for analyzing and interpreting data. These rules provide the foundation for more complex statistical methods and help in making informed decisions based on data. In the next section, we will explore probability distributions, which describe how probabilities are distributed over different outcomes.