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In probability, two events are considered independent if the occurrence of one event does not affect the occurrence of the other. This means that the outcome of Event A has no influence on the outcome of Event B.

For example, if you flip a coin and roll a die, the result of the coin flip (heads or tails) does not change the likelihood of any specific outcome on the die roll (1 through 6). Each event happens without any interference from the other.

In the context of our exercise, gaining more than 40 pounds during pregnancy and giving birth to a baby girl are independent events. The probability of gaining more than 40 pounds (20.1%) does not influence the probability of having a baby girl (49.5%). This independence is a critical concept as it allows us to use a straightforward method for calculating the joint probability of the two events occurring together.

The joint probability of two events is the probability that both events occur at the same time. When dealing with independent events, we can find this probability by multiplying the probabilities of each individual event.

Mathematically, if we have two independent events, Event A and Event B, the joint probability is calculated as:
\[ P(A \text{ and } B) = P(A) \times P(B) \] In simpler terms, this means we take the probability of the first event happening and multiply it by the probability of the second event happening.

In our example, the probability of gaining more than 40 pounds during pregnancy is 0.201, and the probability of having a girl is 0.495. By multiplying these probabilities, we find:
\[ P(A \text{ and } B) = 0.201 \times 0.495 = 0.099495 \] This tells us that the probability of both events occurring together (having a girl and gaining more than 40 pounds) is approximately 9.95%.

Let's delve deeper into the process of our calculation to ensure clarity and understanding.

Step-by-Step Recap:

• First, identify the individual probabilities of each event.
• The probability of gaining more than 40 pounds during pregnancy is given as 0.201.
• The probability of having a baby girl is given as 0.495.

Since we know the events are independent, we use the formula for the joint probability:
\[ P(A \text{ and } B) = P(A) \times P(B) \]

By substituting the values:
\[ P(A \text{ and } B) = 0.201 \times 0.495 = 0.099495 \]
This result shows that approximately 9.95% of pregnancies will result in both gaining more than 40 pounds and having a baby girl.

Understanding these calculations and the principles behind them is essential for mastering the basics of probability and statistics. Such a step-by-step approach ensures that complex concepts are broken down into manageable parts, facilitating learning and application.