[tex]\begin{gathered} \text{ First step, find the probability of each coin.} \\ \text{ Since there are 24 coins in total (7 quarters, 6 dimes, 3 nickels and 8 pennies)} \\ \text{ The probability is } \\ 25c\rightarrow\frac{7}{24} \\ 10c\rightarrow\frac{6}{24} \\ 5c\rightarrow\frac{3}{24} \\ 1c\rightarrow\frac{8}{24} \\ \text{ To get the expected value, multiply the probability of each coin to their respective} \\ \text{value},\text{ and add them all together} \\ E(X)=25c\cdot\frac{7}{24}+10c\cdot\frac{6}{24}+5c\cdot\frac{3}{24}+1c\cdot\frac{8}{24} \\ E(X)=\frac{175}{24}+\frac{5}{2}+\frac{5}{8}+\frac{1}{3} \\ E(X)=\frac{43}{4}\rightarrow10.75c \\ 10.75\text{ cents rounded to the nearest tenth is 10.8 cents} \\ \text{Therefore, the expected value of 10.8 cents.} \end{gathered}[/tex]
