Question

Nellie is a pediatrics nurse who conducts the first neonatal exam on newborn patients. She makes note of the newborns' height, weight, eye color, and heart rate, among other things. Slow heart rate (100 bpm) Brown eyes 2 15 Blue eyes 1 24 Green eyes 2 22 14 5 Grey eyes Are the events "the baby has brown eyes" and "the baby has a fast heart rate (>100 bpm)" independent? no yes

Answer 1

Two events, A and B, are independent if:

$P(A\cap B)=P(A)\cdot P(B)$

Let's define:

A: the baby has brown eyes

B: the baby has a fast heart rate

number of babies with brown eyes: 2 + 15 = 17

number of babies with fast heart rate: 15 + 24 + 22 + 14 = 75

number of babies with brown eyes and a fast heart rate: 15

total number of babies: 2 + 1 + 2 + 5 + 75 = 85

Then:

$\begin{gathered} P(A)=\frac{17}{85} \\ P(B)=\frac{75}{85} \\ P(A\cap B)=\frac{15}{85}=\frac{3}{17} \\ P(A)\cdot P(B)=\frac{17}{85}\cdot\frac{75}{85}=\frac{1275}{7225}=\frac{\frac{1275}{425}}{\frac{7225}{425}}=\frac{3}{17} \end{gathered}$

In conclusion, the events are independent