A building contractor buys 70 of his cement from supplier A

A building contractor buys 70% of his cement from supplier A and 30% from supplier B. A total of 85% of the bags from A arrived undamaged, and a total of 70% of the bags from B arrive undamaged. Find the probability that a damaged bag is from supplier A.

The probability is _______

Solution

Let A be the event that the building contractor buys cement from supplier A.

B be the event that the building contractor buys cement from supplier B.

C be the event of getting a damaged bag.

given that A building contractor buys 70% of his cement from supplier A and 30% from supplier B.

so P[A]=0.70 and P[B]=0.30

A total of 85% of the bags from A arrived undamaged, and a total of 70% of the bags from B arrive undamaged.

hence total percentage of the bags from A arrived damaged is (100-85)%=15%

and total percentage of the bags from B arrived damaged is (100-70)%=30%

so P[C|A]=0.15   and P[C|B]=0.30

hence P[C]=P[C|A]*P[A]+P[C|B]*P[B]=0.15*0.70+0.30*30=0.195   [by theorem of total probability]

hence the probability that a damaged bag is from supplier A is

P[A|C]=P[C|A]*P[A]/P[C]   [by Bayes\' Theorem]

           =0.15*0.70/0.195=0.53846=53.846%

hence the probability is 0.53846   [answer]