Question

Research has shown that 1 women in 400 carry a mutation of the BRCA gene. About 6 out of 10 women with this mutation develop breast cancer. About 1 out of 10 women without this mutation develop breast cancer. (a) Find the probability that a randomly selected woman will carry the mutation of the BRCA gene and will develop breast cancer. Give your answer as a decimal. (b) Are the events “carrying the mutation” and “developing breast cancer” independent? Briefly explain.

Answer 1

P[  women carrying a mutation of the BRCA gene ] = 1/400

P[  women is not carrying a mutation of the BRCA gene ] = 1 - P[  women carrying a mutation of the BRCA gene ]

P[  women is not carrying a mutation of the BRCA gene ] = 1 - 1/400

P[  women is not carrying a mutation of the BRCA gene ] = 399/400

P[ women develop  breast cancer |   women carrying a mutation of the BRCA gene ] = 6/10

P[ women develop  breast cancer |   women is not carrying a mutation of the BRCA gene ] = 1/10

(a) Find the probability that a randomly selected woman will carry the mutation of the BRCA gene and will develop breast cancer.

P[ will carry the mutation of the BRCA gene and will develop breast cancer. ] = P[ women develop  breast cancer |   women carrying a mutation of the BRCA gene ] *P[  women carrying a mutation of the BRCA gene ]

P[ will carry the mutation of the BRCA gene and will develop breast cancer. ] = 6/10*(1/400)

P[ will carry the mutation of the BRCA gene and will develop breast cancer. ] = 6/4000

P[ will carry the mutation of the BRCA gene and will develop breast cancer. ] = 0.0015

(b) Are the events “carrying the mutation” and “developing breast cancer” independent?

if the events are independent.

P[ will carry the mutation of the BRCA gene and will develop breast cancer. ] = P[  women carrying a mutation of the BRCA gene ]* P[ women develop  breast cancer ]

P[ women develop  breast cancer ] = P[ women develop  breast cancer |   women carrying a mutation of the BRCA gene ] *P[  women carrying a mutation of the BRCA gene ] + P[ women develop  breast cancer |   women is not carrying a mutation of the BRCA gene ] *P[  women is not carrying a mutation of the BRCA gene ] =

P[ women develop  breast cancer ] = (6/10)*(1/400) + (1/10)*(399/400)

P[ women develop  breast cancer ] = 6/4000 + 399/4000

P[ women develop  breast cancer ] = 405/4000

P[  women carrying a mutation of the BRCA gene ]* P[ women develop  breast cancer ] = (1/400)*(405/4000)

P[  women carrying a mutation of the BRCA gene ]* P[ women develop  breast cancer ] = 405/1600000

P[  women carrying a mutation of the BRCA gene ]* P[ women develop  breast cancer ] = 0.000253125‬

Which is not equal to P[ will carry the mutation of the BRCA gene and will develop breast cancer. ] = 0.0015

hence the events are not independent