X and Y are independent normally distributed random variables such that X has mean 32 and variance 25, and Y has mean 43 and variance 96. Find

(a) P(X 43),

(b) P(X-Y>0),

(c) P(2X-Y>0). (NEAB) 7. The times taken by two runners A and B to run 400 m races are independent and normally distributed with means 45.0 s and 45.2 s, and standard deviations 0.5 s and 0.8 s respectively. The two runners are to complete in a 400 m race for which there is a track record of 44.5 s.

(a) Calculate, to three decimal places, the probability of runner A breaking the track record.

(b) Show that the probability of runner B breaking the track record is greater than that of runner A.

(c) Calculate, to three decimal places, the probability of runner A beating runner B. (NEAB)