Will It Be a Boy or a Girl?
A pregnant woman often opts to have an ultrasound to predict the gender of her baby.
Assume the following facts are known:
 Fact 1: 48% of the babies born are female.
 Fact 2: The proportion of girls correctly identified is 9 out of 10.
 Fact 3: The proportion of boys correctly identified is 3 out of 4.
(Source: Keeler, Carolyn, and Steinhorst, Kirk. “New Approaches to Learning Probability in the First Statistics Course,” Journal of Statistics Education 9(3):1–24, 2001.)
Here are the questions we want to answer:

Question 1:
If the examination predicts a girl, how likely is it that the baby will be a girl?

Question 2:
If the examination predicts a boy, how likely is it that the baby will be a boy?
Let’s consider what the possibilities are.
 The ultrasound examination predicts a girl, and either (a) a girl is born or (b) a boy is born.
 The ultrasound exam predicts a boy, and either (a) a girl is born or (b) a boy is born.
Let’s represent these four possible outcomes in a twoway table. On the left we have the categorical variable prediction, and on the top the categorical variable gender of baby.
 Girl  Boy  
Predict Girl    
Predict Boy    
   
Now we find ourselves in an interesting situation. A twoway table without data!
The key idea is to create a twoway table consistent with the stated facts, then use the table to answer our questions.
To get started, let’s assume we have ultrasound predictions for 1,000 random babies. We could have picked any number here, but 1,000 will make our calculations easier to keep track of.
Starting with this number, we work backwards with our three facts to fill in this “hypothetical” table.
The first step is to put 1,000 as the overall total in the bottom right corner.
 Girl  Boy  Row Totals 
Predict Girl    
Predict Boy    
Column Totals    1,000 
Let’s consider Fact 1: 48% of the babies born are female.
The bottom row gives the distribution of the categorical variable gender of baby. We can use this fact to compute the total number of girls and boys.
 48% girls means that 0.48 (1,000) = 480 are girls.
 52% are boys. (If 48% are girls, then 100% − 48% = 52% are boys.) So, 0.52(1,000) = 520 boys.
Fill these values into the bottom row of table.
 Note: These are marginal totals.
 You can check your work: These numbers should add to 1,000. If we add all the girls and boys together, we get the total number of babies.
 Girl  Boy  Row Totals 
Predict Girl    
Predict Boy    
Column Totals  0.48(1,000) = 480  0.52(1,000) = 520  1,000 
Now let’s move on to Fact 2: The proportion of girls correctly identified is 9 out of 10.
 9 out of 10 is 90% (9 ÷ 10 = 0.90 = 90%).
 90% of the girls are correctly identified: 0.90(480) = 432.
 10% of the girls are misidentified (predicted to be a boy): 0.10(480) = 48.
Fill these values into the table.
 You can check your work: These numbers should add to the total number of girls.
 (Girls who are correctly identified as girls ) + (Girls who are misidentified as boys) = Total girls
 Girl  Boy  Row Totals 
Predict Girl  0.90(480)= 432   
Predict Boy  0.10(480) = 48   
Column Totals  480  520  1,000 
Finally, we use Fact 3: The proportion of boys correctly identified is 3 out of 4.
 3 out of 4 is 75% (3 ÷ 4 = 0.75 = 75%).
 75% of the boys are correctly identified: 0.75(520) = 390.
 25% of the boys are misidentified (predicted to be a girl): 0.25(520) = 130.
Fill these values into the table.
 You can check your work: These numbers should add to the total number of boys.
 (Boys who are correctly identified as boys ) + (Boys who are misidentified as girls) = Total boys
 Girl  Boy  Row Totals 
Predict Girl  432  0.25(520) = 130  
Predict Boy  48  0.75(520) = 390  
Column Totals  480  520  1,000 
Filling in the Row Totals, we now have a complete hypothetical twoway table based on our given information.
 Girl  Boy  Row Totals 
Predict Girl  432  130  562 
Predict Boy  48  390  438 
Column Totals  480  520  1,000 
We are now in a position to answer our two questions:
Question 1:
If the examination predicts a girl, how likely is it that the baby will be a girl?
Answer: We are asked to find the probability of a girl given that the examination predicts a girl.
This is the conditional probability: P(girl  predict girl).
So our answer to Question 1 is P(girl  predict girl) = 432 / 562 = 0.769.
Question 2:
If the examination predicts a boy, how likely is it that the baby will be a boy?
Answer: We are asked to find the probability of a boy given that the examination predicts a boy.
This is the conditional probability: P(boy  predict boy).
So our answer to Question 2 is P(boy  predict boy) = 390 / 438 = 0.890.
Conclusion: If an ultrasound examination predicts a girl, the prediction is correct about 77% of the time. In contrast, when the prediction is a boy, it is correct 89% of the time.