Bayes' Rule

$4\mathrm{.}1[1$ pt$]$ Suppose a business releases a new product and sets a price of $$4\mathrm{.}25.$ Its customers' valuations for the new product are distributed normally with a mean of $$4\mathrm{.}00$ and standard deviation of $$0\mathrm{.}37.$ What is the probability that one individual customer's valuation of the product is less than $$4\mathrm{.}25?$ What is the probability that one individual customer's valuation of the product is greater than $$4\mathrm{.}25.$ Store your answers in the variables prob$\_$reject and prob$\_$accept.

In $[]$:

prob$\_$reject $=$ None

prob$\_$accept $=$ None

# YOUR CODE HERE

raise NotImplementedError$()$

In $[]$:

#hidden tests from problem $4\mathrm{.}1$ are within this cell

$4\mathrm{.}2[2$ pt$]$ Suppose that the price is still $$4\mathrm{.}25$ and the standard deviation is still $$0\mathrm{.}37,$ but now the business doesn't know if the mean of its customers' valuations is $$3\mathrm{.}95$ or $$4\mathrm{.}05.$ It estimates that the probability of the former is $40\%$ and the probability of the latter is $60\%.$ Then it observes the decision of one of its customers about whether or not to purchase the product. If the customer doesn't purchase the product, what would be the updated probability that the mean of the customers' valuations is $$3\mathrm{.}95?$ Store your answer in the variable prob$\_395\_$if$\_$reject. If the customer purchases the product, what would be the updated probability that the mean of the customers' valuations is $$3\mathrm{.}95?$ Store your answer in the variable prob$\_395\_$if$\_$accept.

$In[]$:

$pro{b}_{395}{?}_i{f}_r$eject $=$ None

$pro{b}_{395}{?}_i{f}_a$ccept $=$ None

$\frac{?}{?}$#YOUR CODE HERE

raise NotImplementedError$()$