Question

According to a study published by a group of University of Massachusetts sociologists, approximately 60% of the Valium users in the state of Massachusetts first took Valium for psychological problems. Find the probability that among the next 8 users from this state who are interviewed, (a) exactly 3 began taking Valium for psychological problems;(b) at least 5 began taking Valium for problems that were not psychological.

Answer 1

Answer:

(a) 0.124

(b) 0.174

Step-by-step explanation:

We are given that 60% of the Valium users in the state of Massachusetts first took Valium for psychological problems.

The Binomial distribution probability is given by;

P(X = r) = $\binom{n}{r}p^{r}(1-p)^{n-r}$ for x = 0,1,2,3,.......

Here, n = number of trials(samples) which is 8 in our case

           r = no. of success

          p = probability of success which is probability of users who take  

                 Valium for psychological problems of 0.60 in our case

(a) Let X = users taking Valium for psychological problems

  So, P(X = 3) = $\binom{8}{3}0.6^{3}(1-0.6)^{8-3}$

                      = $56 * 0.6^{3}*0.4^{5}$ = 0.124

(b) Since, it is given that 60% of the Valium users in the state of  Massachusetts first took Valium for psychological problems which means 40% of the Valium users in the state of Massachusetts take Valium for problems which are not psychological.

i.e., in this case p = 0.40

So, P(X >= 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)

= $\binom{8}{5}0.4^{5}(1-0.4)^{8-5} + \binom{8}{6}0.4^{6}(1-0.4)^{8-6} + \binom{8}{7}0.4^{7}(1-0.4)^{8-7} + \binom{8}{8}0.4^{8}(1-0.4)^{8-8}$

= $56 * 0.4^{5} * (0.6)^{3} + 28 * 0.4^{6} * (0.6)^{2} + 8 * 0.4^{7} * (0.6)^{1} + 1 * 0.4^{8}$

= 0.174