Problem 13 - Algebra 2

Question

Suppose \(A\) and \(B\) are independent events, with \(P(A)=0.60\) and \(P(B)=0.25\) Find each probability. a. \(P(A \text { and } B)\) b. \(P(A | B)\) c. What do you notice about \(P(A)\) and \(P(A | B) ?\) d. Critical Thinking One way to describe \(A\) and \(B\) as independent events is The occurrence of \(B\) has no effect on the probability of \(A .\) Explain how the answer to part (c) illustrates this relationship.

Step-by-Step Solution

Answer

a. \(P(A \text { and } B) = 0.15\) \n b. \(P(A | B) = 0.60\)\n c. The probabilities \(P(A)\) and \(P(A | B)\) are equal, which indicates that B's occurrence does not affect the odds of A happening.\n d. This equality illustrates the principle of independence, where the occurrence of one event does not affect the possibility of another event.

Step 1: Identify Given Probabilities

From the problem, we know that \(P(A) = 0.60\) and \(P(B) = 0.25\).

Step 2: Calculate P(A and B)

For independent events A and B, the probability of both A and B occurring is given by \(P(A \text{ and } B) = P(A) \cdot P(B)\), which gives us \(0.60 \cdot 0.25 = 0.15\)

Step 3: Calculate Conditional Probability P(A | B)

For independent events A and B, the conditional probability of A given B is equal to the probability of A. Therefore, \(P(A | B) = P(A) = 0.60\).

Step 4: Observe Relationship between P(A) and P(A | B)

From above, we see that \(P(A) = P(A | B)\), indicating the probabilities stay the same even when we know that B has happened.

Step 5: Explain Relationship

The fact that \(P(A) = P(A | B)\) is the very definition of independence. This means that the occurrence of event B does not change the probability of event A, hence they are independent events.

Key Concepts

Probability

Probability is a fundamental concept in statistics and mathematics that deals with the likelihood of an event occurring.
It ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty. In simpler terms, probability tells you how likely something is to happen.

In the context of the given exercise, two events, A and B, are considered. We know:

• The probability of event A occurring, denoted as \(P(A)\), is 0.60.
• The probability of event B occurring, \(P(B)\), is 0.25.

These probabilities express the chance of each event happening independently of each other. They form the basis for calculating other probabilities, such as the probability that both events occur together.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred.
This is represented as \(P(A | B)\), which stands for 'the probability of A given B'.

In the scenario where events A and B are independent, the conditional probability of A given B simplifies to \(P(A | B) = P(A)\).
This is because the occurrence of B does not affect the likelihood of A happening. So, in our example,
we calculate:

• \(P(A | B) = P(A) = 0.60\)

This shows that knowing B happened doesn't change the probability of A happening.
It beautifully illustrates the independence of the two events, affirming that the occurrence of one does not provide any information about the occurrence of the other.

Multiplication Rule for Probability

The multiplication rule for probability is used when determining the probability of two independent events occurring simultaneously.
This involves multiplying the probabilities of each event.
For independent events A and B, this rule is expressed as:

• \(P(A \text{ and } B) = P(A) \cdot P(B)\)

Using the given probabilities in the exercise:

• \(P(A) = 0.60\)
• \(P(B) = 0.25\)

Thus, the probability of both A and B occurring, \(P(A \text{ and } B)\), is calculated as:

• \(0.60 \cdot 0.25 = 0.15\)

This means there is a 0.15 chance that both events occur together, underscoring how independent probabilities are handled.
The multiplication rule clearly shows the independence by maintaining a straightforward product calculation when events do not influence each other.