553.420/620 Probability
Assignment #01
Due FEB-02 by 11:59pm as a PDF upload to Canvas Gradescope.
1. A coin is flipped 10 times.
(a) How many sequences are there total?
(b) How many sequences start with a head?
(c) How many sequences start and end with the same flip?
(d) How many sequences have all of flips the same?
(e) How many sequences have exactly one head in the first 3 flips? (f) How many sequences have exactly one head?
(g) How many sequences have exactly one head in both the first 5 and last 5 flips? (h) How many sequences alternate in flips?
(i) How many sequences have the first 5 flips all the same face and last 5 flips all the same face?
(j) * How many sequences have less than 9 heads?
(k) * How many sequences have at least one head and at least one tail?
(l) ** How many sequences have exactly one run of exactly 4 heads?
* better to think of the complement. ** little challenging.
2. Alfred picks shirts, shorts, and 2 shoes for his outfit. Alfred has 7 shirts, 6 shorts, and 4 pairs of shoes. The 7 shirts are the colors of the rainbow (Red, Orange, Yellow, Green, Blue, Indigo, Violet). The 6 shorts are all of the colors of the rainbow excluding violet. The 4 pairs of shoes are red, yellow, green, and blue. An outfit is a selection of shirt, shorts and 2 shoes. WARNING: There are 4 matching pairs of shoes, and 4 are left and 4 are right. A selection of shoes is a selection of 2 shoes from the 8 (not necessarily a match).
(a) How many distinct outfits can Alfred make? No restrictions. Note: shoes don’t have to match.
(b) How many distinct outfits can Alfred make having a matching pair of shoes?
(c) How many distinct outfits can Alfred make having a left and right shoe?
(d) How many ways can Alfred make an outfit if he wants to wear the same color on all of his pieces?
(e) How many ways can Alfred make an outfit if he wants each of his pieces to be a different color? Assume he picks a pair of shoes.
(f) How many outfits have his shirt blue? Assume a pair of shoes are taken.
(g) How many outfits have his shirt blue? Assume that a left and right shoe are taken.
(h) How many outfits where he doesn’t wear orange? Assume a pair of shoes are taken.
(i) How many outfits where he doesn’t wear red? Assume a pair of shoes are taken.
(j) How many outfits where he doesn’t wear red? Assume a left and right shoe are taken.
(k) How many outfits have his shirt and shorts the same color? Assume a pair of shoes are taken.
(l) How many outfits have exactly one garment of clothing orange? Assume a pair of shoes are taken. In this question, the pair of shoes will be considered as one garment of clothing.
(m) How many outfits have exactly one garment of clothing red? Assume a pair of shoes are taken. In this question, the pair of shoes will be considered as one garment of clothing.
(n) How many outfits have exactly one garment of clothing orange? Assume that a left and right shoe are taken out. In this question, each shoe is considered as one garment of clothing.
(o) How many outfits have exactly one garment of clothing is red? Assume that a left and right shoe are taken out. In this question, each shoe is considered as one garment of clothing.