553.420/620 Probability
Assignment #02
Due FEB-09 by 11:59pm as a PDF upload to Canvas Gradescope.
1. (a) How many 3-digit integers can be formed using the digits 0 , 1, 2, 3, 4, 5, 6, 7, 8, and 9, where each digit is used at most once.
(b) How many of these 3-digit integers in part (a) are odd?
(c) How many of these 3-digit integers in part (a) are (strictly) greater than 330?
Remark: In part (a) the hundreds-digit cannot be 0 else you’d have a 2-digit integer. Be careful in parts (b) and (c), for example, in part (c) break the problem into two pieces: those integers that start with a 3 and those that don’t.
2. Consider the word PRESSURE (note there are two R’s, two E’s and 2 S’s in this word). (a) How many anagrams of this word are possible?
(b) How many anagrams start and end with an S?
(c) How many anagrams have no two E’s next to each other? Try to count this two distinct ways.
3. Consider the six (6) letters: a,b,c,d,e,f. Let m and k be integers that satisfy m ≥ 2 and 1 ≤ k ≤ 6. A game is played where m people are asked to select any k of these six. After a person selects they put the letters back for the next person. People do not see others’ selections. The experimenter observes the selections of each person. You may assume the selections are unordered. (a) How many possible sample points are there in this experiment?
(b1) If m = 2 and k = 3, compute the probability that all selections are disjoint.
(b2) If m = 2 and k = 3, compute the probability that there is a common element in all selections and no two selections have any other elements in common.
(c1) Repeat (b1) using m = 3 and k = 2.
(c2) Repeat (b2) using m = 3 and k = 2.
4. We have 16 m&m’s of which 4 are red, and the remaining 12 are non-red. Out of these 16 we are going to give 4 to each of 4 people.
(a) What’s the probability that person 1 gets only red m&m’s (i.e., no other color)? (b) What’s the probability that each person ends up with exactly one red m&m?
(c) What’s the probability that person 1 and 2 combined have a total of (exactly) 2 red m&m’s?
5. There are n = 30 chips numbered 1 through 30 in a hat. Each of k = 7 people reaches into the hat, draws a chip, and records the chip number. Then they return the chip to the hat for the next draw. Let A be the event that all k people record distinct numbers.
(a) Determine the probability of A. Leave it as an expression involving the n and k in this problem – no need to reduce it yet.
(b) Strictly speaking, Ac is the event that not all k people record distinct numbers. In the context of this problem that means that a chip that was selected was selected again by someone else, i.e., there is a repeated number in the list, equivalently, more than one person recorded the same number. The complementary rule of probability states P (Ac ) = 1 - P (A). I ask that you use a calculating device (computer, calculator, etc.) to estimate P (Ac ) to 6-digits beyond the decimal. FYI: This probability is a little shocking to me!
Remark: When n = 365 and k is arbitrary, this is called The birthday problem: in a room with k = 23 people there is better than a 50% chance that there is a matching birthday, and when k = 41 there is better than a 90% chance.
6. Consider the resulting polynomial when (2a - b)30 is multiplied out. Without multiplying this thing out, determine the coefficient of the term a3 b27 and simplify it. Hint: (2a - b)30 = (x + y)30 with x = 2a and y = -b.