Problem 16 - A First Course in Probability

Question

Ninety-eight percent of all babies survive delivery. However, 15 percent of all births involve Cesarean (C) sections, and when a C section is performed, the baby survives 96 percent of the time. If a randomly chosen pregnant woman does not have a C section, what is the probability that her baby survives?

Step-by-Step Solution

Answer

The probability that the baby survives if the woman does not have a C section is approximately 98.35%.

Step 1: Identify the events

We need to identify the events we are interested in. Let A be the event that the baby survives and B be the event that the woman has a C section. We are looking for the probability of the baby surviving given that there is no C section, or P(A | B').

Step 2: Find the probabilities of events

We have been given: - P(A) = 0.98 (The probability of a baby surviving) - P(B) = 0.15 (The probability of a C section) - P(A | B) = 0.96 (The probability of a baby surviving given there's a C section) Now, we have to find the probability of a baby surviving given that there is no C section, or P(A | B').

Step 3: Apply the conditional probability formula

The conditional probability formula is: \(P(A | B') = \frac{P(A \cap B')}{P(B')}\) First, we need to find P(B'), which is the probability of not having a C section: P(B') = 1 - P(B) = 1 - 0.15 = 0.85 Next, we can find P(A ∩ B') using the formula: \(P(A \cap B') = P(A) - P(A \cap B)\) We can find P(A ∩ B) using the given conditional probability P(A | B): \(P(A \cap B) = P(A | B) * P(B) = 0.96 * 0.15 = 0.144\) Now we can find P(A ∩ B'): \(P(A \cap B') = P(A) - P(A \cap B) = 0.98 - 0.144 = 0.836\)

Step 4: Calculate the probability of surviving given no C section

We can now put all these values to calculate the probability of surviving given no C section: \(P(A | B') = \frac{P(A \cap B')}{P(B')} = \frac{0.836}{0.85} ≈ 0.9835\) So, the probability that the baby survives if the woman does not have a C section is approximately 98.35%.

Key Concepts

Events in Probability

In probability, an event is essentially a set of outcomes to which we assign a probability. Events in probability can either be simple or compound. A simple event involves one possible outcome, while a compound event involves a combination of outcomes.
For example, in the 'baby survival' scenario, several events are at play:

• Event A: The baby survives.
• Event B: A Cesarean (C) section occurs.
• Event B': A natural birth occurs (no C section).

Understanding these events is crucial, as it forms the basis for computing probabilities and determining their relationships and dependencies with respect to one another. Identifying the correct events helps in applying the right probability techniques to solve problems.

Probability Formula

To solve probability problems, it is essential to make use of probability formulas which include conditional probability, multiplication rules, and the addition rule.
The Conditional Probability Formula, for instance, is used to find the likelihood of event A happening given that event B has happened. It is expressed as:\[P(A | B) = \frac{P(A \cap B)}{P(B)}\]
In simpler terms, we calculate the intersection of events A and B (both events happening together) and divide it by the probability of event B occurring. Applying this to our exercise, we aim to find P(A | B'), the probability of the baby surviving when there is no C section. This requires understanding and calculating the individual probabilities before using the formula.

Bayes' Theorem

Bayes' Theorem is a concept in probability that helps us update our beliefs about the probability of a hypothesis based on new evidence. Although not explicitly required in the exercise, Bayes' Theorem provides a framework for understanding conditional probabilities.
It states:\[P(B | A) = \frac{P(A | B) \cdot P(B)}{P(A)}\]
This formula allows us to find the probability of event B given that event A has happened, using the probability of A given B, the probability of B, and the probability of A. While solving the given exercise does not demand Bayes' Theorem directly, the comprehension of how probabilities update with new information enriches our understanding of conditional probability.

Probability Distribution

Probability distribution provides a comprehensive picture of all possible outcomes of a random variable and the likelihood of each outcome. It can be discrete, with specific values, or continuous for a range of values.
In real-world scenarios like baby deliveries, we often deal with discrete distributions since we have defined outcomes, such as survival versus non-survival. Calculating distributions involves:

• Listing all possible outcomes.
• Assigning and summing probabilities to ensure they add up to 1.
• Utilizing formulas and theorems to find specific probabilities.

Understanding probability distributions helps us visualize and handle uncertainties effectively, as seen in evaluating baby survival rates under different childbirth methods.