Problem 90 - A First Course in Probability

Question

Suppose that \(n\) independent trials, each of which results in any of the outcomes \(0,1,\) or \(2,\) with respective probabilities \(p_{0}, p_{1},\) and \(p_{2}, \sum_{i=0}^{2} p_{i}=1\) are performed. Find the probability that outcomes 1 and 2 both occur at least once.

Step-by-Step Solution

Answer

The probability that outcomes 1 and 2 both occur at least once in \(n\) independent trials is: \[P(\text{1 and 2 occur at least once}) = 1 - [(p_0 + p_2)^n + (p_0 + p_1)^n - p_0^n]\]

Step 1: Find the probability of the complementary events

To begin, let's first find the probability that outcome 1 never occurs. In this case, the only possible outcomes for each trial are 0 and 2. The probability of this happening in a single trial is \(p_0 + p_2\). Since we have \(n\) independent trials, we raise this probability to the power of n: \((p_0 + p_2)^n\). Similarly, the probability that outcome 2 never occurs is \((p_0 + p_1)^n\). Finally, the probability that both outcome 1 and outcome 2 never occur is \(p_0^n\).

Step 2: Apply complementary event principle and sum probabilities

Now, let's use the complementary event principle. The probability that outcome 1 or outcome 2 never occur is the sum of the individual probabilities, minus the probability that both outcome 1 and outcome 2 never occur (that is, the intersection of the events): \[P(\text{complementary}) = (p_0 + p_2)^n + (p_0 + p_1)^n - p_0^n\]

Step 3: Calculate the desired probability

Finally, to find the probability that both outcomes 1 and 2 occur at least once, we will subtract the probability of the complementary event from 1: \[P(\text{1 and 2 occur at least once}) = 1 - P(\text{complementary}) = 1 - [(p_0 + p_2)^n + (p_0 + p_1)^n - p_0^n]\] The probability that outcomes 1 and 2 both occur at least once in \(n\) independent trials is: \[P(\text{1 and 2 occur at least once}) = 1 - [(p_0 + p_2)^n + (p_0 + p_1)^n - p_0^n]\]

Key Concepts

Complementary Events

In probability theory, complementary events refer to pairs of dichotomous outcomes where the occurrence of one event implies the non-occurrence of the other and vice versa. For example, if you flip a coin, the result could either be heads or tails; these are complementary events. If one happens, the other can't.

When dealing with complementary events, it might be easier to calculate the probability of an event by using its complement. Because the sum of probabilities for all possible outcomes equals 1, the probability of an event happening is 1 minus the probability of it not happening.

In the given problem, we calculated the probability of the complementary event, which is when outcomes 1 and 2 do not occur at least once. This involves calculating each of the individual probabilities of not having 1 or not having 2, and subtracting the overlap (probability of neither 1 nor 2).

This approach simplifies calculations and avoids the direct computation of increasingly complex scenarios, especially when independent trials are involved.

Independent Trials

Independent trials represent situations where the outcome of one trial doesn't affect the outcome of another. Each trial is like hitting reset; no matter what happened before, the odds remain the same for each new attempt.

In this problem, the assumption is that each trial, which can result in outcome 0, 1, or 2, is independent. This independence allows us to multiply probabilities when calculating the likelihood of events over multiple trials.

• To find the probability of a specific sequence of outcomes, multiply their individual probabilities.
• In series of trials, if the probability for one outcome stays constant, it's an indication of independence.

This concept plays a crucial role in computing probabilities across many scenarios, ensuring that cumulative calculations account for repeated, unchanged conditions.

Conditional Probability

Conditional probability is used to find the likelihood of an event occurring, given that another event has already happened. It's how probability adapts when we're informed by additional data.

Mathematically, conditional probability, where we want to find the probability of event A given event B has occurred, is usually written as:\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
In our exercise, conditional probability isn't directly applied, but understanding it helps assimilate complementary events. It provides insight into how probabilities adjust when certain conditions are established. Though our problem is solved by complementarity and independence, knowing when events aren't independent would lead us to use this concept, changing how we calculate related probabilities. Understanding these foundational tools can expand your reasoning in probability and enhance problem-solving skills efficiently.