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. \(P\left(S_{2} \mid A\right)\)

Short answer

Answer: The conditional probability of event \(S_2\) occurring, given that event \(A\) is observed, is approximately \(0.3846\).

Step-by-step solution

Understand Bayes' Theorem

Bayes' theorem is a formula that allows us to find the probability of an event given some other events, using the conditional probability and the probability of those related events. The formula for Bayes' theorem is: $$ P(S_i | A) = \frac{P(A | S_i) \cdot P(S_i)}{P(A)} $$ In our case, we want to compute \(P(S_2 | A)\), and we are given the values of \(P(A | S_2)\) and \(P(S_2)\).

Calculate the probability of event A

To apply Bayes' theorem, we need the probability of event \(A\). We can find this using the law of total probability, which states: $$ P(A) = \sum_{i=1}^n P(A | S_i) \cdot P(S_i) $$ In our case, n=3, and we have the necessary information to compute this sum: $$ P(A) = P(A | S_1) \cdot P(S_1) + P(A | S_2) \cdot P(S_2) + P(A | S_3) \cdot P(S_3) = 0.2 \cdot 0.2 + 0.1 \cdot 0.5 + 0.3 \cdot 0.3 = 0.13 $$

Apply Bayes' theorem to find \(P(S_2 | A)\)

Now that we have the probability of event \(A\), we can use Bayes' theorem to find \(P(S_2 | A)\): $$ P(S_2 | A) = \frac{P(A | S_2) \cdot P(S_2)}{P(A)} = \frac{0.1 \cdot 0.5}{0.13} \approx 0.3846 $$ Hence, the probability of event \(S_2\) occurring, given that event \(A\) is observed, is approximately \(0.3846\).

Key concepts

Conditional Probability

Conditional probability is a measure of the likelihood of an event occurring given that another event has already occurred. It's an essential concept in probability and statistics that allows you to update the probabilities of events based on new information.

For example, suppose you want to find the probability of drawing a red card from a standard deck of playing cards. The probability of drawing a red card is 1/2. However, if you know that you've drawn a card from only hearts and diamonds (the red suits), the conditional probability of having a red card given this new information is 1. Conversely, if you know the card is from clubs or spades, the conditional probability of it being red is 0.

In the exercise example, the probability of event A occurring given that event S2 has occurred, denoted as \(P(A | S_{2})\), is used to better understand the likelihood of A when S2 is known to have happened. This concept is what Bayes' theorem leverages to deduce unknown probabilities.

Law of Total Probability

The law of total probability is a fundamental rule that relates marginal probabilities to conditional probabilities. It states that the probability of an event can be found by considering all possible scenarios that could lead to that event.

To calculate the total probability of an event A, you sum the probabilities of A occurring in the context of each scenario (Si). You can picture this as breaking down A into a series of mutually exclusive events that cover all possible outcomes:

• \(P(A) = P(A | S_1) \times P(S_1) + P(A | S_2) \times P(S_2) + P(A | S_3) \times P(S_3)\)

In the textbook problem, we have three mutually exclusive events S1, S2, and S3. The law tells us how likely event A is in total, considering each of these scenarios and their specific likelihoods of leading to A. This calculation is crucial because it gives us the denominator needed in Bayes' theorem to update our beliefs about the probability of each scenario Si given that A has occurred.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. In other words, the occurrence of one event excludes the possibility of another event occurring simultaneously. An intuitive example is flipping a coin; it can't land as both heads and tails on the same flip.

Applying this to our exercise, the events S1, S2, and S3 are mutually exclusive. This means they represent distinct outcomes of an experiment, and only one can happen at any given time. When calculating probabilities, this exclusivity simplifies the process because it ensures that there is no overlap between the events.

Here's an important point about mutually exclusive events: When you're dealing with them, their intersection (the probability of both events occurring together) is always zero. This means that the law of total probability becomes particularly straightforward since you don't need to subtract any overlapping probabilities, unlike when dealing with non-mutually exclusive events.