Answer:
The probability that a student passed the test given that they did not complete the assignment is
.
Step-by-step explanation:
The probability of an event <em>E</em> is the ratio of the number of favorable outcomes <em>n</em> (E) to the total number of outcomes <em>N</em>.
The union of two events is:
![P(A\cup B)=P(A)+P(B)-P(A\cap B)](https://tex.z-dn.net/?f=P%28A%5Ccup%20B%29%3DP%28A%29%2BP%28B%29-P%28A%5Ccap%20B%29)
The intersection of the complements of two events is:
![P(A^{c}\cap B^{c})=1-P(A\cup B)](https://tex.z-dn.net/?f=P%28A%5E%7Bc%7D%5Ccap%20B%5E%7Bc%7D%29%3D1-P%28A%5Ccup%20B%29)
The condition probability of an event given that another event has already occurred is:
![P(B|A)=\frac{P(A\cap B)}{P(A)}](https://tex.z-dn.net/?f=P%28B%7CA%29%3D%5Cfrac%7BP%28A%5Ccap%20B%29%7D%7BP%28A%29%7D)
Denote the events as follows:
<em>A</em> = students who passed the test
<em>B</em> = students who completed the assignment
Given:
N = 27
n (A) = 17
n (B) = 22
= 3
Compute the value of P (<em>A</em> ∪ <em>B</em>) as follows:
![P(A^{c}\cap B^{c})=1-P(A\cup B)](https://tex.z-dn.net/?f=P%28A%5E%7Bc%7D%5Ccap%20B%5E%7Bc%7D%29%3D1-P%28A%5Ccup%20B%29)
![P(A\cup B)=1-P(A^{c}\cap B^{c})](https://tex.z-dn.net/?f=P%28A%5Ccup%20B%29%3D1-P%28A%5E%7Bc%7D%5Ccap%20B%5E%7Bc%7D%29)
![=1-\frac{3}{27}\\](https://tex.z-dn.net/?f=%3D1-%5Cfrac%7B3%7D%7B27%7D%5C%5C)
![=\frac{24}{27}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B24%7D%7B27%7D)
Compute the value of P (A ∩ B) as follows:
![P(A\cup B)=P(A)+P(B)-P(A\cap B)](https://tex.z-dn.net/?f=P%28A%5Ccup%20B%29%3DP%28A%29%2BP%28B%29-P%28A%5Ccap%20B%29)
![P(A\cap B)=P(A)+P(B)-P(A\cup B)](https://tex.z-dn.net/?f=P%28A%5Ccap%20B%29%3DP%28A%29%2BP%28B%29-P%28A%5Ccup%20B%29)
![=\frac{17}{27}+\frac{22}{27}-\frac{24}{27}\\](https://tex.z-dn.net/?f=%3D%5Cfrac%7B17%7D%7B27%7D%2B%5Cfrac%7B22%7D%7B27%7D-%5Cfrac%7B24%7D%7B27%7D%5C%5C)
![=\frac{17+22-24}{27}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B17%2B22-24%7D%7B27%7D)
![=\frac{15}{27}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B15%7D%7B27%7D)
Compute the value of P (A | B) as follows:
![P(A|B)=\frac{P(A\cap B)}{P(B)}](https://tex.z-dn.net/?f=P%28A%7CB%29%3D%5Cfrac%7BP%28A%5Ccap%20B%29%7D%7BP%28B%29%7D)
![=\frac{15/27}{22/27}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B15%2F27%7D%7B22%2F27%7D)
![=\frac{15}{22}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B15%7D%7B22%7D)
Thus, the probability that a student passed the test given that they did not complete the assignment is
.