Question

7 at a local community college, students are required to take two introductory english courses, english 101 and english 102. english 101 must be completed first. out of the students in english 101, 40% earn an a. of the students who earn an a in english 101, 30% go on to earn an a in english 102. of the students who do not earn an a in english 101, 10% earn an a in english 102. let ( x = ) the number of a’s earned by a randomly selected student in the introductory english courses. which of the following tables shows the probability distribution of ( x )?
a) (\begin{array}{c|c|c|c}x & 0 & 1 & 2 \\ hline p(x) & \frac{1}{3} & \frac{1}{3} & \frac{1}{3}end{array})
b) (\begin{array}{c|c|c|c}x & 0 & 1 & 2 \\ hline p(x) & 0.42 & 0.08 & 0.50end{array})
c) (\begin{array}{c|c|c|c}x & 0 & 1 & 2 \\ hline p(x) & \frac{1}{5} & \frac{2}{5} & \frac{2}{5}end{array})
d) (\begin{array}{c|c|c|c}x & 0 & 1 & 2 \\ hline p(x) & 0.54 & 0.34 & 0.12end{array})
e) (\begin{array}{c|c|c|c}x & 0 & 1 & 2 \\ hline p(x) & 0.20 & 0.10 & 0.70end{array})

Explanation:

Step1: Calculate $P(X=2)$

Probability of A in both courses: $0.4 \times 0.3 = 0.12$

Step2: Calculate $P(X=0)$

Probability of no A in either: $(1-0.4) \times (1-0.1) = 0.6 \times 0.9 = 0.54$

Step3: Calculate $P(X=1)$

Subtract sum of $P(X=0)$ and $P(X=2)$ from 1: $1 - 0.54 - 0.12 = 0.34$

Step4: Match to options

The probabilities $0.54, 0.34, 0.12$ match option D.

Answer:

D)

$$\begin{array}{c|ccc} X & 0 & 1 & 2 \\ \hline P(X) & 0.54 & 0.34 & 0.12 \end{array}$$