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Understanding how to calculate probability is essential when determining the likelihood of a particular event happening. Probability is the measure of how likely something is to occur, ranging from 0 (impossible) to 1 (certain). To calculate probability, especially in scenarios like our battery example, we can follow a simple approach:

• Identify the total number of possible outcomes.
• Determine the number of favorable outcomes.
• Divide the number of favorable outcomes by the total number of possible outcomes.

This forms the basic formula for probability: \[ P = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} \]
In more complex problems, like our flashlight example, understanding the sequence and combination of independent outcomes (like batteries having a 90% chance) will guide you to use the appropriate method—such as binomial distribution, which we'll talk about next.

When you're dealing with situations where a fixed number of independent experiments (trials) are conducted, such as checking if a flashlight works, the binomial probability formula becomes useful. This formula helps to calculate the probability of having a certain number of successful outcomes in a series of independent events. Let's break it down:1. **Parameters**: - **n**: the total number of trials or experiments (in our case, 10 flashlights). - **k**: the number of successful trials (either 9 or 10 flashlights working). - **p**: the probability of success on a single trial (0.81 for a flashlight working).The formula is expressed as:\[ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} \]Where:- \( \binom{n}{k} \) symbolizes combinations and refers to the number of ways to choose \( k \) successes from \( n \) trials.- \( p^k \): probability of having exactly \( k \) successes.- \( (1-p)^{n-k} \): probability of having \( n-k \) failures.
For our flashlight example, applying this formula enables us to calculate the chances of exactly 9 or 10 flashlights working, helping us sum those probabilities to determine the overall likelihood of at least 9 working flashlights.

In probability theory, independent events are crucial because the outcome of one event does not affect the outcome of another. This assumption simplifies calculations significantly and is fundamental to applying methods like the binomial probability formula.When we say events are independent:- The occurrence of one does not change the likelihood of the other occurring.- Mathematically, two events A and B are independent if: \[ P(A \cap B) = P(A) \times P(B) \]
In the flashlight exercise, the acceptable voltage of each battery is considered an independent event. This means that whether a battery is good or not does not impact the status of another. The independence allows us to multiply probabilities for serial events—such as calculating that a flashlight works only if both batteries do. Hence, the probability of a flashlight working is derived by multiplying the probability of each battery having acceptable voltage (0.9 for each), leading to the result \( (0.9)^2 \ = 0.81 \), establishing a foundation for further probability calculations.