As a manufacturer of a flu vaccine, you must decide how many doses of vaccinations to make. You know the following probability distribution describes the distribution of demand for vaccination: there is a \( 42 \% \) chance demand will be 10 million, a \( 20 \% \) chance that demand will be 14 million claim, a \( 17 \% \) chance that demand will be 19 million, and otherwise, demand will be 24 million. What is the expected value of demand for vaccinations? (please express your answer in millions and use 2 decimal places, eg, if your answer is \( 11,560,000 \) then please write that as 11.560 )

To find the expected value of the demand for vaccinations, you can use the formula for expected value: \[ E(X) = \sum (x_i \cdot P(x_i)) \] Where \( x_i \) is the demand and \( P(x_i) \) is the probability of that demand. Given data: - \(10\) million with a probability of \(0.42\) - \(14\) million with a probability of \(0.20\) - \(19\) million with a probability of \(0.17\) - \(24\) million with a probability of \(0.01\) (since the remaining probability, \(0.01\) implies it is \(1 - (0.42 + 0.20 + 0.17) = 0.01\)) Calculating the expected value: \[ E(X) = (10 \cdot 0.42) + (14 \cdot 0.20) + (19 \cdot 0.17) + (24 \cdot 0.01) \] \[ = 4.20 + 2.80 + 3.23 + 0.24 \] \[ = 10.47 \] Thus, the expected value of demand for vaccinations is **10.470** million.