The molarity of a solute in solution is defined to be the number of moles of solute per liter of solution \(\left(1ight.\) mole \(=6.02 \times 10^{23}\) molecules \()\). If \(X\) is the molarity of a solution of sodium chloride \((\mathrm{NaCl})\), and \(Y\) is the molarity of a solution of sodium carbonate \(\left(\mathrm{Na}_{2} \mathrm{CO}_{3}ight)\), the molarity of sodium ion \(\left(\mathrm{Na}^{+}ight)\)in a solution made of equal parts \(\mathrm{NaCl}\) and \(\mathrm{Na}_{2} \mathrm{CO}_{3}\) is given by \(M=0.5 X+Y\). Assume \(X\) and \(Y\) are independent and normally distributed, and that \(X\) has mean 0.450 and standard deviation 0.050 , and \(Y\) has mean 0.250 and standard deviation 0.025 .

a. What is the distribution of \(M\) ?

b. Find \(P(M>0.5)\).