Dependent and Independent Events

Two events A and B are considered dependent events if the probability of one event occurring p(A) affects the probability of the other event occurring p(B). If there is no such influence, they are known as independent events.

Understanding whether events are independent or dependent is crucial because the method for calculating the joint probability of two events differs based on this distinction.

• Independent Events
  For independent events, to find the probability of both events occurring p(A∩B), you multiply their individual probabilities p(A) and p(B). $$ P(A \cap B) = P(A) \cdot P(B) $$

  Example: Tossing a coin and rolling a die are independent events because the result of the coin toss does not affect the outcome of the die roll, and vice versa.

• Dependent Events
  For dependent events, to calculate the probability of both events occurring p(A∩B), you multiply the probability of event A by the conditional probability of B given A. $$ P(A \cap B) = P(A) \cdot P(B|A) $$

  Example: If it rains (event A), the likelihood of someone carrying an umbrella (event B) increases. This is a classic example of dependent events.

In general, the product rule applies to both independent and dependent events when calculating probabilities.

This is because the conditional probability p(B|A) is equivalent to the probability p(B) of event B when A and B are independent.

Independent Events

Two events are considered independent events if the occurrence of one does not affect the probability of the other. Mathematically, events A and B are independent if $$ P(A \cap B) = P(A) \cdot P(B) $$ where P denotes probability.

In other words, A and B are independent if the probability of both happening, P(A∩B)—the probability of the intersection of A and B—is the same as the product of their individual probabilities, P(A) and P(B).

$$ P(A∩B)=P(A) \cdot P(B) $$

With independent events, calculating probability is simple: you just multiply the probabilities of each event.

A Practical Example

Imagine flipping a coin twice, aiming to get "heads" both times.

There are two favorable outcomes:

• Event A: "Heads on the first flip".
• Event B: "Heads on the second flip".

The probability of event A is p(A)=1/2 since the coin has two sides, and only one is "heads".

$$ P(A) = \frac{1}{2} $$

The probability of event B is also p(B)=1/2.

$$ P(B) = \frac{1}{2} $$

These events are independent because the outcome of the second flip (p(B)) isn't influenced by the outcome of the first flip.

So, to find the probability of getting "heads" both times, you multiply the probabilities of the individual events.

$$ P(A∩B)=P(A) \cdot P(B) $$

$$ P(A∩B)= \frac{1}{2} \cdot \frac{1}{2} $$

$$ P(A∩B)= \frac{1}{4} $$

$$ P(A∩B)= 0.25 $$

The probability of both events occurring is P(A∩B)= 0.25, or 25%. In other words, one out of every four attempts will result in "heads" both times.

Dependent Events

Two events are considered dependent events if the probability of one event affects the probability of the other. For dependent events, we use conditional probability p(B|A). $$ P(A \cap B) = P(A) \cdot P(B|A) $$

Typically, events are dependent when there is some kind of causal or conditional connection between them.

Note: If events are independent (rather than dependent), the conditional probability p(B|A) equals p(B). Therefore, the calculation becomes the same as for independent events. $$ P(A \cap B) = P(A) \cdot P(B|A) = P(A) \cdot p(B) $$

A Practical Example

Imagine you draw two cards from a deck of 52, aiming to draw two aces.

Consider these two events:

• Event A: "Drawing an ace on the first draw".
• Event B: "Drawing an ace on the second draw".

The probability of event A (drawing an ace on the first draw) is P(A)=4/52 since there are 4 aces in a deck of 52 cards.

$$ P(A)= \frac{4}{52} $$

After the first draw, only 51 cards remain in the deck.

Now, the probability of event B (drawing an ace on the second draw) depends on the outcome of event A:

• If event A occurs, then an ace was drawn on the first try, leaving 3 aces in a deck of 51 cards. In this case, the probability of event B is p(B)=3/51 $$ P(B)= \frac{3}{51} = 0.058 $$
• If event A does NOT occur, then no ace was drawn on the first try, leaving 4 aces in a deck of 51 cards. Here, the probability of event B is p(B)=4/51 $$ P(B)= \frac{4}{51} = 0.078 $$

So, the probability of drawing an ace on the second draw (event B) directly depends on the result of the first draw (event A).

This is a clear example of dependent events.

In such situations, to calculate the joint probability, you use the conditional probability p(B|A), assuming that the first event has already occurred.

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

The probability of the first event is p(A)=4/52, while the conditional probability of the second event is p(B|A)=3/51, as it depends on the first card being an ace.

$$ P(A \cap B) = \frac{4}{52} \cdot \frac{3}{51} = 0.0045 $$

In summary, understanding the difference between dependent and independent events is essential for accurately assessing probabilities and making well-informed decisions in uncertain situations.