The National Institute of Mental Health reports that there is a 20% chance of an adult American suffering from a psychiatric disorder. Four randomly selected adult Americans are examined for psychiatric disorders.
a. If you let a success correspond to an adult American having a psychiatric disorder, what is the success probability, p? (The use of the word success in Bernoulli trials need not reflect its usually positive connotation.)
b. Construct a table similar to Table 5.14 on page 228 for the four people examined. Display the probabilities to four decimal places.
c. Draw a tree diagram for this problem similar to the one shown in Fig. 5.4 on page 228.

Table 5.14

Figure 5.4

d. List the outcomes in which exactly three of the four people examined  ave a psychiatric disorder.

e. Find the probability of each outcome in part (d). Why are those probabilities all the same?

f. Use parts (d) and (e) to determine the probability that exactly three of the four people examined have a psychiatric disorder.

g. Without using the binomial probability formula, obtain the probability distribution of the random variable Y , the number of adults out of four who have a psychiatric disorder.

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Probability Outcome (0.8)(0.8)(0.8) =0.512 (0.8)(0.8)(0.2) = 0.128 (0.8)(0.2)(0.8) = 0.128 (0.8)(0.2)(0.2) = 0.032 (0.2)(0.8)(0.8) =0.128 (0.2)(0.8)(0.2) =0.032 (0.2)(0.2)(0.8) = 0.032 (0.2)(0.2)(0.2) = 0.008 SSS ssf sfs sff fss fsf ffs ff First Second Third person person person Outcome Probability SS (0.8)(0.8)(0.8) = 0.512 0.8 0.2 0.8 of ssf (0.8)(0.8)(0.2) = 0.128 0.2 sfs (0.8)(0.2)(0.8) = 0.128 0.8 0.8 0.2 (0.8)(0.2)(0.2) = 0.032 of sff fss (0.2)(0.8)(0.8) = 0.128 0.8 0.2 0.8 0.2. fsf (0.2)(0.8)(0.2) = 0.032 0.2 ffs (0.2)(0.2)(0.8) = 0.032 0.8 0.2 (0.2)(0.2)(0.2) = 0.008 fff