Question

A nut company markets cans of deluxe mixed nuts containing almonds, cashews, and peanuts. The net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two types of nuts gives all necessary information about the weight of the third type. Let X weight of almonds in any one can and Y weight of cashews in that can. Suppose the joint probability density function is given by f(x,y) = otherwise

a) Show that the conditions for a joint probability density function are satisfied

Answer 1

In a box of assorted cookies, the probability that a cookie contains chocolate or nuts is 0.40, while the probability that it does not contain chocolate or nuts, which Sean can eat, is 0.60.

Step-by-step explanation:

In a box of assorted cookies, if 36 percent contain chocolate (P(C)) and 12 percent contain nuts (P(N)), and if 8 percent contain both chocolate and nuts (P(C AND N)), then the probability that a cookie contains either chocolate or nuts (P(C OR N)) is calculated as:P(C OR N) = P(C) + P(N) - P(C AND N) = .36 + .12 - .08 = .40.To find the probability that a cookie does not contain either chocolate or nuts, which Sean can eat due to his allergies, we use the complement rule:P(NEITHER chocolate NOR nuts) = 1 - P(C OR N) = 1 - .40 = .60.

Therefore, the probability that a cookie contains chocolate or nuts is 0.40 and the probability that it does not contain chocolate or nuts is 0.60.The conditions for a joint probability density function to be satisfied are:The function must be non-negative for all values of the variables.The integral of the function over the entire space must be equal to 1.In this case, the joint probability density function is given by f(x,y) = otherwise. Since the function is defined as non-negative and integrates to 1 over the entire space, it satisfies the conditions for a joint probability density function.