Q6 Problem 5

10 Points

For this problem, leave your answers as fractions or mathematical expressions with fractions, e.g., (3/4 + 1/2)

You have two buckets of candy before you. The candy comes in three types: fruity, chocolatey, and healthy.

For the fruity candy, there are Gummy Bears, Sour Patch Kids, and Twin Snakes. For the chocolatey candy, there are KitKats, Reeses, and Snickers. For the healthy candy there are Granola Gratin, Carob Crunch, and Bran Bites.

In Bucket 1: 7 gummy bears 4 twin snakes 4 KitKats 8 Reeses 12 Granola Gratin 4 Bran Bites

In Bucket 2: 5 Twin Snakes 10 Sour Patch Kids 5 Snickers 5 KitKats 7 Carob Crunch 8 Bran Bites 4 Granola Gratin

You flip a biased coin to decide which bucket to choose from. If it is heads you choose bucket 1, if tails bucket 2. The coin shows tails 60 percent of the time.

(PART I) A good candy is chocolatey or fruity. You flip the coin, then choose a candy at random from the corresponding bucket. What is the probability you got a good candy?

(PART II) Assume that we have the same set up as Part I, with the same candy counts in all the buckets. Once again, you flip the biased coin to determine the bucket, and then you pick a candy at random. The candy is a chocolatey candy. What is the probability it came from the first bucket?

(PART III) Once again consider the same set up with the same candy counts. You again flip the biased coin to determine the bucket. You then select a candy at random from the corresponding bucket. (You do not replace the candy.) You then flip the coin a second time to determine the bucket and then select another candy at random from that bucket. What is the expected number of candies from the first bucket after these two draws? Your final answer cannot have a summation symbol