Answer to Question #341448 in Statistics and Probability for Queen

Question

Research has shown that 55% of new Small Medium Enterprises (SMEs) are started by graduates while 45% are started by non-graduates. It is also known that 70% of SMEs started by graduates are successful i.e they suvived beyond 3 years, while only 10% of those started by non-graduates are successful.

a. What is the probability that a new SME is successfull

b. What is the probability that a new SME is successful and it was not started by a graduate

c. If it is know that a new SME is successful, what is the probability that it was not started by a graduatre

Answer 1

Let $A$ denote the event "SME is started by graduate".

Let $B$ denote the event "SME is started by non-graduate".

Let $S$ denote the event "SME is successful".

a.

$$P(S)=P(A)P(S|A)+P(B)P(S|B)$$

$$=0.55(0.7)+0.45(0.1)=0.43$$

b.

$$P(S\cap B)=P(B)P(S|B)=0.45(0.1)=0.045$$

c.

$$P(B|S)=\dfrac{P(B)P(S|B)}{P(A)P(S|A)+P(B)P(S|B)}$$

$$=\dfrac{0.045}{0.43}\approx0.10465$$