Lesson 2.2: The Laws of Probability

Introduction

In this lesson, we will explore the fundamental laws of probability that are essential in statistical analysis. The learning objectives for this lesson include:

Understanding the addition law for the union of events, including mutually exclusive events.
Understanding the multiplication law, including cases of independent events.
Calculating and comparing single, independent, and mutually exclusive probabilities.
Applying the addition law to find the probability of a union of events.
Applying the multiplication law to find the probability of combinations of events.

To engage your interest, consider this scenario: In a card game, you can either draw a red card or a face card. What are the chances of drawing a card that is either red or a face card? This question will require us to use the laws of probability.

The Addition Law of Probability

The addition law of probability helps us calculate the probability of the union of two events. The formal statement of this law is:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Mutually Exclusive Events

If events $ A $ and $ B $ cannot occur simultaneously, they are referred to as mutually exclusive events. For mutually exclusive events, the addition law simplifies:

$$P(A \cup B) = P(A) + P(B)$$

Example 1: Mutually Exclusive Events

Suppose we have a standard 52-card deck. Let event $ A $ be drawing a red card, and event $ B $ be drawing a face card. These events are not mutually exclusive, as there are red face cards. Thus, we will first calculate $ P(A) $, $ P(B) $, and $ P(A \cap B) $.

Probability of drawing a red card, $ P(A) = \frac{26}{52} = \frac{1}{2} $ (since there are 26 red cards).
Probability of drawing a face card, $ P(B) = \frac{12}{52} = \frac{3}{13} $ (since there are 12 face cards).
Probability of drawing a red face card, $ P(A \cap B) = \frac{6}{52} = \frac{3}{26} $ (there are 3 red face cards).

Now, using the addition law:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Substituting the values:

$$P(A \cup B) = \frac{1}{2} + \frac{3}{13} - \frac{3}{26}$$

To perform the calculation, we need a common denominator. The least common multiple of 2, 13, and 26 is 26.

Convert $ \frac{1}{2} $: $ \frac{1 \cdot 13}{2 \cdot 13} = \frac{13}{26} $
Convert $ \frac{3}{13} $: $ \frac{3 \cdot 2}{13 \cdot 2} = \frac{6}{26} $

Now substituting:

$$P(A \cup B) = \frac{13}{26} + \frac{6}{26} - \frac{3}{26} = \frac{16}{26} = \frac{8}{13}$$

Non-Mutually Exclusive Events

In cases where two events can occur simultaneously, we must consider the probability of their intersection to avoid double counting.

Example 2: Non-Mutually Exclusive Events

Let’s consider the same card scenario above with $ A $ and $ B $. However, $ A $ is now defined as drawing a King, and $ B $ is still joining the face card. Now we calculate:

Probability of drawing a King, $ P(A) = \frac{4}{52} = \frac{1}{13} $
Probability of a face card remains unchanged, $ P(B) = \frac{12}{52} = \frac{3}{13} $
Probability of drawing a King that is also a face card, $ P(A \cap B) = \frac{4}{52} = \frac{1}{13} $

Using the addition rule:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Substituting:

$$P(A \cup B) = \frac{1}{13} + \frac{3}{13} - \frac{1}{13}$$

$$= \frac{3}{13}$$

The Multiplication Law of Probability

The multiplication law is used to find the probability of two events happening in sequence. This law is defined as follows:

$$P(A \cap B) = P(A) \cdot P(B|A)$$

Where $ P(B|A) $ is the conditional probability of event $ B $ occurring given that event $ A $ has occurred.

Independent Events

If two events $ A $ and $ B $ are independent, the outcome of one does not affect the other. In this case, the multiplication law can be simplified:

$$P(A \cap B) = P(A) \cdot P(B)$$

Example 3: Independent Events

Consider a fair six-sided die. Let event $ A $ be rolling a 3, and event $ B $ be rolling an even number. The probabilities are:

$ P(A) = \frac{1}{6} $ (only one 3 on a die).
$ P(B) = \frac{3}{6} = \frac{1}{2} $ (numbers 2, 4, 6 are even).

These events are independent because the outcome of rolling a die does not affect whether we roll an even number. Therefore, we can find:

$$P(A \cap B) = P(A) \cdot P(B) = \frac{1}{6} \cdot \frac{1}{2} = \frac{1}{12}$$

Dependent Events

If two events influence each other, they are dependent. In such cases, we need to use the conditional probability:

Example 4: Dependent Events

Imagine drawing two cards from a deck without replacement. Let $ A $ be the first card drawn is a heart, and $ B $ is the second card drawn is also a heart.

Probability of drawing a heart first, $ P(A) = \frac{13}{52} = \frac{1}{4} $
After drawing one heart, there are now 12 hearts and 51 total cards.
Probability of drawing a heart second given the first was a heart: $ P(B|A) = \frac{12}{51} $

Using the multiplication law:

$$P(A \cap B) = P(A) \cdot P(B|A) = \frac{1}{4} \cdot \frac{12}{51} = \frac{12}{204} = \frac{3}{51}$$

Conclusion

In this lesson, we covered the laws of probability, focusing on the addition law for the union of events and the multiplication law for combinations of events. We explored both mutually exclusive and non-mutually exclusive events using well-documented examples. Furthermore, we differentiated between independent and dependent events when applying the multiplication law. Mastering these concepts is fundamental to progressing in the broader subject of statistics.

Study Notes

The addition law: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
Application to mutually exclusive events: $$P(A \cup B) = P(A) + P(B)$$
The multiplication law for independent events: $$P(A \cap B) = P(A) \cdot P(B)$$
The multiplication law for dependent events: $$P(A \cap B) = P(A) \cdot P(B|A)$$
Important to distinguish between mutually exclusive and independent events when calculating probabilities.