Question

You want to learn how to make cakes. You have ten ingredients to use in your cake, all contained in bottles. Two of them contain baking soda, three contain flour, and 5 contain brown sugar. You are going to keep baking cakes all day till you make a successful one. To create a successful cake you have to use a tablespoon of the baking soda. Each time you attempt to make a cake, you mix all of the other ingredients (not the baking soda) and then randomly pick a bottle from the 10 and use a tablespoon of the ingredient you have picked. If it was the baking soda bottle, your cake will be a success and you can stop right there. If it is any other ingredient, you need to start all over and make another and another till you have a successful cake.

An attempt using baking-soda will take you 1 hour and 30 minutes to mix and bake.

An attempt using flour takes you 1 hour and 15 minutes to mix and bake.

An attempt with brown sugar takes you 3 hours to mix and bake.

What is the expected amount of time it will take you to successfully make your cake?

If you can explain yourself that would be helpful.

Answer 1

We can calculate the probability to get each bottle:

Baking soda = 2/10 = 0.2

Flour = 3/10 = 0.3

Brown sugar =5/10 = 0.5

X(Time taken)	P(X)	X*P(X)
90	0.20	18.000
75	0.30	22.500
180	0.50	90.000

mean = E[X] = Σx*P(X) =            130.5 minutes

Expected time = 2 hour 10.5 minutes.

Please revert in case of any doubt.

Please upvote. Thanks in advance