Compound Probability
This chapter discusses further concepts that lie at the core of probability theory.
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Compound probability deals with the likelihood of multiple events occurring together or in sequence. Unlike simple events, which involve a single outcome like flipping one coin, compound events involve multiple outcomes such as flipping two coins or rolling two dice. Understanding this distinction is fundamental to probability theory, as it allows us to analyze more complex scenarios where multiple random processes interact.
For independent events, we use the multiplication rule: the probability of both events occurring equals the product of their individual probabilities. When flipping two coins, the probability of getting heads on both is one-half times one-half, which equals one-fourth. Similarly, when rolling two dice, the probability of getting six on both dice is one-sixth times one-sixth, which equals one thirty-sixth. This rule applies because the outcome of one event does not affect the outcome of the other.
When events are dependent, the occurrence of one event affects the probability of another. We use conditional probability notation P of B given A to represent this relationship. For dependent events, the probability of both A and B occurring equals P of A times P of B given A. Consider drawing two cards without replacement: the probability of drawing two kings is four fifty-seconds times three fifty-firsts, since after drawing the first king, only three kings remain out of fifty-one cards.
The addition rule helps us find the probability of either one event or another occurring. For mutually exclusive events that cannot happen simultaneously, we simply add their probabilities. However, for non-mutually exclusive events that can overlap, we must subtract the probability of both events occurring to avoid double counting. For example, when drawing a card that is either a king or a heart, we add the probability of drawing a king plus the probability of drawing a heart, then subtract the probability of drawing the king of hearts, since it was counted in both categories.
Complex probability trees help us solve multi-stage problems involving both dependent and independent events. Consider a medical test scenario: a disease affects one percent of the population, and the test is ninety-five percent accurate. Using a probability tree, we can calculate that even with a positive test result, the actual probability of having the disease is only about sixteen percent. This demonstrates how probability trees systematically organize complex calculations and reveal counterintuitive results in real-world applications.