[20 points] Suppose there are three options to treat a child's throat infection - ask the physician for the antibiotic without performing a lab test; conduct a lab test and then give antibiotics only if the result is positive; or do nothing. The probability of the child's throat being infected is 50%. An antibiotic costs $10, whereas a lab test costs $5. If the child receives an antibiotic without being infected, they may develop a resistance to antibiotics with probability 0.2 and costs $30 for further treatment. If the child is infected and receives the antibiotic, they will be cured immediately without any side effects. If the child is infected but not treated, the infection will become a disease costing $1000. (a) [5 points] Draw the decision tree for the child's parents with these scenarios. (The decision tree is similar to the Gamble tree from notes/textbook when a patient is presented with different options for treating a particular disease). (b) [10 points] Calculate the expected cost of each possible outcome: Doing nothing, getting antibiotics without performing a lab test, getting antibiotics after performing a lab test (c) [5 points] To minimize cost, what should be the parents' optimal decision