Question

When an experiment is conducted, one and only one of three mutually exclusive events \(S_{1}, S_{2}\) and \(S_{3}\), can occur, with \(P\left(S_{1}\right)=.2, P\left(S_{2}\right)=.5,\) and \(P\left(S_{3}\right)=.3 .\) The probabilities that an event A occurs, given that event \(S_{1}, S_{2}\), or \(S_{3}\) has occurred are $$ P\left(A \mid S_{1}\right)=.2 \quad P\left(A \mid S_{2}\right)=.1 \quad P\left(A \mid S_{3}\right)=.3 $$ If event A is observed, use this information to find the probabilities in Exercises 4 -6. \(P\left(S_{1} \mid A\right)\)

Short answer

Answer: The probability is \(\frac{2}{9}\).

Step-by-step solution

Calculate the probability of event A

Using the Law of Total Probability: $$P(A) = P(A \mid S_1) \cdot P(S_1) + P(A \mid S_2) \cdot P(S_2) + P(A \mid S_3) \cdot P(S_3)$$ $$P(A) = (0.2)(0.2) + (0.1)(0.5) + (0.3)(0.3)$$

Simplify the result

Perform the multiplications and sums: $$P(A) = 0.04 + 0.05 + 0.09$$ $$P(A) = 0.18$$

Use Bayes' Theorem

Input the values into Bayes' Theorem to find \(P(S_1 \mid A)\): $$P(S_1 \mid A) = \frac{P(A \mid S_1) \cdot P(S_1)}{P(A)}$$ $$P(S_1 \mid A) = \frac{(0.2)(0.2)}{0.18}$$

Simplify the result

Perform the multiplication and division: $$P(S_1 \mid A) = \frac{0.04}{0.18}$$

Write the final answer as a fraction or decimal

Simplify the fraction if possible: $$P(S_1 \mid A) = \frac{2}{9}$$ So, the probability of event \(S_1\) occurring, given that event A has occurred, is \(\frac{2}{9}\).

Key concepts

Law of Total Probability

The Law of Total Probability is a fundamental concept that helps in calculating the probability of an event occurring, considering several mutually exclusive scenarios. When dealing with a problem involving multiple events, which individually cover all possible outcomes, this law becomes extremely useful. Suppose an event \( A \) can occur in conjunction with other events like \( S_1, S_2, \) and \( S_3 \), which are mutually exclusive and exhaustive. The probability of \( A \) happening, given these scenarios, is calculated by multiplying the probability of each scenario with its conditional probability and adding them together:
\[ P(A) = P(A \mid S_1) \cdot P(S_1) + P(A \mid S_2) \cdot P(S_2) + P(A \mid S_3) \cdot P(S_3) \]
This equation sums up the probabilities across all possible ways \( A \) can occur. In the problem provided, event \( A \) is associated with three exclusive events \( S_1, S_2, \) and \( S_3 \), with specified probabilities. Applying this law helps in understanding how these individual probabilities contribute to the overall probability of the event \( A \).

Conditional Probability

Conditional probability refers to the likelihood of an event occurring based on the occurrence of another event. It is expressed as \( P(A \mid B) \), indicating the probability of event \( A \) occurring given that event \( B \) has happened. In this exercise, the conditional probabilities are given as follows:

• \( P(A \mid S_1) = 0.2 \)
• \( P(A \mid S_2) = 0.1 \)
• \( P(A \mid S_3) = 0.3 \)

These values denote how likely event \( A \) is to occur in the context of each specific scenario. For instance, if \( S_1 \) occurs, the chance of \( A \) happening is 0.2 or 20%. Here's where it gets interesting - these probabilities play a crucial role in determining overall and conditional outcomes using Bayes' Theorem later. Essentially, conditional probabilities allow us to focus on a particular slice of the probability landscape detailed enough to draw meaningful insights for each condition provided.

Probability Calculation

Calculating probabilities can involve a variety of approaches, depending on the information available. The exercise combines different concepts to find the conditional probability of \( S_1 \) if \( A \) has occurred using Bayes' Theorem. Here’s a concise step-by-step breakdown:

• First, ascertain total occurrence probability of \( A \) using the Law of Total Probability. This was calculated as 0.18.
• Then, apply Bayes' Theorem to find \( P(S_1 \mid A) \). The theorem expresses this probability as:
  \[ P(S_1 \mid A) = \frac{P(A \mid S_1) \cdot P(S_1)}{P(A)} \]
• Substituting the values, calculate as:
  \[ P(S_1 \mid A) = \frac{0.04}{0.18} = \frac{2}{9} \]

Thus, the exercise concludes with a fractional answer, representing how Bayes' Theorem efficiently uses known probabilities to solve for unknown probabilities. Specific steps like multiplication and simplification help students identify key probability values step by step, confirming comprehensive understanding of probability calculations.