Complements and Special Cases

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Calculating a probability directly can sometimes be very complicated, especially in situations involving multiple outcomes or conditions like "at least one" or "not all". In these cases, it is often easier to use the complement rule. The complement rule states:

The probability of an event happening is equal to 1 minus the probability that it does not happen.
Mathematically, $P(event) = 1 - \overline{P(event)}$

Using the complement simplifies the calculation because the complement often involves just one straightforward case, while the original problem may involve many different possibilities that are harder to count or compute directly.

Using the Complements for "At Least One"

A student flips a fair coin 5 times and wants to know the probability of getting at least one heads. Calculating this directly would require adding up the probabilities of getting 1, 2, 3, 4, or 5 heads, which involves several separate computations. Instead, it's much easier to find the probability of the complement (getting no heads at all), which means all 5 flips are tails.

$P(\text{at least one head}) = 1 - P(\text{no heads})$

The probability of tails on one flip is 0.5, so the probability of getting five tails is $(0.5)^5$.

$P(\text{at least one head} = 1 - (0.5)^5$
$P(\text{at least one head}) = 1 - 0.03125$
$P(\text{at least one head}) = 0.96875$

Using the Complements for "Not All"

A factory produces 10 light bulbs, and the probability that any single bulb is defective is 0.1. If someone randomly selects 3 bulbs, what is the probability that not all of them are defective? Finding the probability directly would involve calculating the probabilities for 0, 1, or 2 defective bulbs, which can be messy. Using the complement, we calculate the probability that all three are defective: