Is ultrasound a reliable method for determining the gender of an unborn baby? There is accompanying data on 1,000 births in the table below.
Ultrasound
Predicted
Female	Ultrasound
Predicted
Male
Actual Gender Is
Female	439	47
Actual Gender Is
Male	132	382
(a)
Use the given information to estimate the probability that a newborn baby is female, given that the ultrasound predicted the baby would be female. (Round your answer to three decimal places.)
 
Incorrect: Your answer is incorrect.
(b)
Use the given information to estimate the probability that a newborn baby is male, given that the ultrasound predicted the baby would be male. (Round your answer to three decimal places.)
 
Incorrect: Your answer is incorrect.

We can solve these problems using **conditional probability**. Specifically, we're given the ultrasound predictions and actual gender outcomes, and we need to find the probabilities of the actual gender given the prediction.

### (a) Probability that the baby is female given the ultrasound predicted female

The formula for conditional probability is:

\[
P(\text{Actual Female} | \text{Predicted Female}) = \frac{P(\text{Actual Female and Predicted Female})}{P(\text{Predicted Female})}
\]

From the table:
- \( P(\text{Actual Female and Predicted Female}) = 439 \) (this is where the actual gender is female and the ultrasound predicted female).
- \( P(\text{Predicted Female}) = 439 + 132 = 571 \) (this is the total number of cases where the ultrasound predicted female, whether correct or incorrect).

Now, we compute:

\[
P(\text{Actual Female} | \text{Predicted Female}) = \frac{439}{571} \approx 0.769
\]

### (b) Probability that the baby is male given the ultrasound predicted male

Similarly, for the probability that the baby is male given that the ultrasound predicted male:

\[
P(\text{Actual Male} | \text{Predicted Male}) = \frac{P(\text{Actual Male and Predicted Male})}{P(\text{Predicted Male})}
\]

From the table:
- \( P(\text{Actual Male and Predicted Male}) = 382 \) (this is where the actual gender is male and the ultrasound predicted male).
- \( P(\text{Predicted Male}) = 47 + 382 = 429 \) (this is the total number of cases where the ultrasound predicted male, whether correct or incorrect).

Now, we compute:

\[
P(\text{Actual Male} | \text{Predicted Male}) = \frac{382}{429} \approx 0.890
\]

### Final Answers:
- (a) The probability that the baby is female given the ultrasound predicted female is approximately **0.769**.
- (b) The probability that the baby is male given the ultrasound predicted male is approximately **0.890**.

Would you like further details on these calculations or have any questions?

Here are 5 related questions to expand on this:
1. What is the probability that the ultrasound incorrectly predicted a female?
2. How can the accuracy of ultrasound predictions be improved?
3. What is the total accuracy of the ultrasound predictions based on the data?
4. How would you calculate the probability that the ultrasound predicts the wrong gender?
5. How does this method compare with other gender prediction methods?

**Tip:** Conditional probability helps determine how likely an event is based on certain given information, making it useful in scenarios like this one with medical predictions.

Use the given information to estimate the probability that a newborn baby is female, given that the ultrasound predicted the baby would be female. Also, estimate the probability that a newborn baby is male, given that the ultrasound predicted the baby would be male.

Learn how to use conditional probability to estimate the likelihood of a newborn being female or male, based on ultrasound predictions. This problem explores ultrasound data and uses Bayes' Theorem to compute the probabilities of correct predictions. Ideal for high school students studying advanced probability topics.

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