91Ó°ÊÓ

In probability, events are considered independent when the occurrence of one event does not affect the occurrence of another. Think of independent events like two separate coin tosses. The result of the first toss doesn't impact the result of the second.
In the context of Sam and Maria taking their driver's test, their outcomes are independent because one passing or failing doesn't influence the other's outcome. It's important to identify such independent events correctly, as it allows us to calculate the combined probability by simply multiplying their individual probabilities.
For Sam and Maria, each has an independent probability of passing the driving test, which is 0.08. Therefore, the probability of both passing is computed as:

• \( P(\text{both pass}) = P(\text{Sam passes}) \times P(\text{Maria passes}) = 0.08 \times 0.08 = 0.0064 \)

By understanding independence, you can easily determine the chances of multiple events happening together without one event influencing the other.

Complementary events encompass all possible outcomes, with a sum probability of 1. For instance, if one event is the likelihood of something occurring, its complement is the likelihood that it doesn't occur. These two probabilities together always equal 1.
In our example, the probability of failing the test is 0.92, while passing is 0.08. These values are complements because:

• \( P(\text{failing}) + P(\text{passing}) = 0.92 + 0.08 = 1 \)

Using complementary events simplifies problems since knowing one event's probability instantly reveals the other's. When calculating at least one of Sam or Maria passing the test, it is easier to find the probability that both fail and subtract that from 1:

• \( P(\text{both fail}) = 0.92 \times 0.92 = 0.8464 \)

Thus:

• \( P(\text{at least one passes}) = 1 - P(\text{both fail}) = 1 - 0.8464 = 0.1536 \)

By utilizing complementary events, one can efficiently calculate probabilities for scenarios involving multiple outcomes.

The probability of passing a test can tell us how likely it is for someone to succeed on their attempt. In Sam and Maria's scenario, we focus on the percentage of teenagers who pass on their first try, which is 8% or 0.08 in decimal form.
While the probability itself seems straightforward, understanding its role is crucial when dealing with combined events. When considering joint probabilities for Sam and Maria, or even calculating the chance of at least one of them passing, we must work with this foundational probability of 0.08.
The stepwise approach involved:

• Multiplying probabilities for joint outcomes: When calculating the probability that both pass, their individual chances (each 0.08) are multiplied together to yield a combined likelihood (0.0064).
• Utilizing complementary probabilities: For at least one passing, first calculate the complement (probability of none passing) and subtract it from 1 to get 0.1536.

Thus, the probability of passing, while initially simple, becomes a critical factor when assessing more complex scenarios.