The above is the chapter 3 exercise 23, please solve the below question

Suppose that it is beliered that p~, the proportion of consumers (in a large population) who will purchase a certain product is either .2, .3, .4, or .5. Furthermore, the prior probabilities for these four values are P(.2)=.2, P(.3)=.3,P(.4)=.3, and P(.5)=.2. A sample of size 10 is taken from the population, and of the 10 consumers, 3 state that they will buy the procluct. What are the posterior probabilities? In calculating the posterior probabilities, what assumptions did you need to make? In Exercise 23 in Chapter 3 , suppose that the prior distribution could be represented by a beta distribution with r=4 and n=10. Find the posterior distribution. Also, find the posterior distribution corresponding to the following beta prior distributions: r=2,n=5;r=8,n=20;r=6,n=15. In each of the four prior distributions considered in this exercise, the mean of the prior distribution is 40 . How, then, do you explain the differences in the means of the posterior distributions