Answer to Question #208966 in Statistics and Probability for mimi

Question

Students taking a module have a year mark above 75 with probability 20%, a year mark between

30 and 75 with probability 60%, and a year mark below 30 with probability 20%. A student with a

year mark above 75 passes the exam with probability 70%, a student with a year mark between 30

and 75 passes the exam with probability 65% and a student with a year mark below 30 passes the

exam with probability 15%.

(a) Construct a probability tree for the described scenario. (6)

(b) What is the probability that a randomly chosen student passes the exam? (6)

(c) If a student has a year mark below 75, what is the probability that the student passes the

exam? (4)

Answer 1

(a) Let $A$ denotes the event "a year mark is above 75".

Let $B$ denotes the event "a year mark is between 30 and 75".

Let $C$ denotes the event "a year mark is below 30 ".

Given $P(A)=0.2, P(B)=0.6, P(C)=0.2.$

Let $Y$ denotes the event "a student passes the exam".

Given $P(Y|A)=0.70, P(Y|B)=0.65, P(Y|C)=0.15.$

(b) The Law of Total Probability

$$P(Y)=P(A)P(Y|A)+P(B)P(Y|B)+P(C)P(Y|C)$$

$$=0.2(0.70)+0.6(0.65)+0.2(0.15)=0.56$$

(c)

$$P(Y|B\cup C)=P(B)P(Y|B)+P(C)P(Y|C)$$

$$=0.6(0.65)+0.2(0.15)=0.42$$