## In a math class with 27 students, a test was given the same day that an assignment was due. There were 17 students who passed the test and 2

Question

In a math class with 27 students, a test was given the same day that an assignment was due. There were 17 students who passed the test and 22 students who completed the assignment. There were 3 students who failed the test and also did not complete the assignment. What is the probability that a student passed the test given that they did not complete the homework

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3 months 2022-02-02T13:45:11+00:00 2 Answers 0 views 0

The probability that a student passed the test given that they did not complete the assignment is $$\frac{15}{22}$$.

Step-by-step explanation:

The probability of an event E is the ratio of the number of favorable outcomes n (E) to the total number of outcomes N.

$$P(E)=\frac{n(E)}{N}$$

The union of two events is:

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

The intersection of the complements of two events is:

$$P(A^{c}\cap B^{c})=1-P(A\cup B)$$

The condition probability of an event given that another event has already occurred is:

$$P(B|A)=\frac{P(A\cap B)}{P(A)}$$

Denote the events as follows:

A = students who passed the test

B = students who completed the assignment

Given:

N = 27

n (A) = 17

n (B) = 22

$$n(A^{c}\cap B^{c})$$ = 3

Compute the value of P (AB) as follows:

$$P(A^{c}\cap B^{c})=1-P(A\cup B)$$

$$P(A\cup B)=1-P(A^{c}\cap B^{c})$$

$$=1-\frac{3}{27}\\$$

$$=\frac{24}{27}$$

Compute the value of P (A ∩ B) as follows:

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

$$P(A\cap B)=P(A)+P(B)-P(A\cup B)$$

$$=\frac{17}{27}+\frac{22}{27}-\frac{24}{27}\\$$

$$=\frac{17+22-24}{27}$$

$$=\frac{15}{27}$$

Compute the value of P (A | B) as follows:

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$

$$=\frac{15/27}{22/27}$$

$$=\frac{15}{22}$$

Thus, the probability that a student passed the test given that they did not complete the assignment is $$\frac{15}{22}$$.