Question

activity 2: genetic testing
a suspected cause of a disease is a variation in a certain gene. a study gathers at - risk people at random and
tests them for the disease as well as for the genetic variation.

	has the disease	does not have the disease	total
does not have the genetic variation	1,055	1,160	2,215
total	1,135	1,172	2,307

a person from the study is selected at random. let a represent the event \has the disease\ and b represent
\has the genetic variation.\

1. use the table to find the probabilities. show your reasoning.

a. ( p(a) )
b. ( p(b) )
c. ( p(a \text{ and } b) )
d. ( p(a|b) )

Explanation:

Part a: \( P(A) \)

Step 1: Identify total and favorable

Total people: \( 2307 \). Favorable (has disease, \( A \)): \( 1135 \).

Step 2: Apply probability formula

Probability \( P(A) = \frac{\text{Number with } A}{\text{Total}} = \frac{1135}{2307} \approx 0.492 \) (simplify or keep as is).

Part b: \( P(B) \)

Step 1: Identify total and favorable

Total people: \( 2307 \). Favorable (has genetic variation, \( B \)): \( 92 \).

Step 2: Apply probability formula

Probability \( P(B) = \frac{\text{Number with } B}{\text{Total}} = \frac{92}{2307} \approx 0.0399 \) (simplify or keep as is).

Part c: \( P(A \text{ and } B) \)

Step 1: Identify intersection

People with both \( A \) (disease) and \( B \) (genetic variation): \( 80 \).

Step 2: Apply probability formula

Probability \( P(A \text{ and } B) = \frac{\text{Number with both}}{\text{Total}} = \frac{80}{2307} \approx 0.0347 \).

Part d: \( P(A|B) \)

Answer:

Step 1: Recall conditional probability

\( P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \) (or \( \frac{\text{Both}}{\text{Total with } B} \)).

Step 2: Substitute values

From table, total with \( B \): \( 92 \), both: \( 80 \). So \( P(A|B) = \frac{80}{92} \approx 0.8696 \).

Final Answers:

a. \( \boldsymbol{\frac{1135}{2307} \approx 0.492} \)
b. \( \boldsymbol{\frac{92}{2307} \approx 0.040} \)
c. \( \boldsymbol{\frac{80}{2307} \approx 0.0347} \)
d. \( \boldsymbol{\frac{80}{92} \approx 0.870} \)