代写553.421 Intro. to Probability honors 2022 Assignment #2代做迭代

日期：2025-02-17 01:03

553.421 Intro. to Probability honors

Assignment #2

Due Friday, Sep. 16 11:59PM as a PDF upload to Gradescope.

2.1. Suppose 1 ≤ k ≤ n are integers.

(a) Show that

(b) Use part (a) to show

2.2. Simplify

2.3. We deal out the 13 cards to each of 4 bridge players (North, South, East, West). What is the probability that North receives 6 spades, South receives 5 spades, and East and West each have 1 spade?

2.4. We roll a fair 6-sided die 12 times. What’s the probability we see each face of the die twice?

2.5. Gary is creating a workout. The order of the exercises he performs is irrelevant. Out of the 28 machines, in how many ways can he select 4 machines to do each day of the week with no repeats?

2.6. A middle row on a plane seats 7 people. Three of them order chicken and the remaining four pasta. The flight attendant returns with the meals, but has forgotten who ordered what and discovers that they are all asleep, so she puts the meals in front of them at random. What is the probability that they all receive correct meals?

2.7. (a) In how many ways can you give 9 children 14 chocolate chip cookies?

(b) Re-do part (a) so that Rick, one of the kids, receives exactly one cookie.

(c) Re-do part (a) so that Rick receives at least two cookies.

(d) Re-do part (a) so that no child goes hungry (i.e., each receives at least one cookie).

2.8. Consider all possible 40-long sequences of the digits 1, 2, 3 and 4. Assuming that these sequences are each equally likely compute the probability a sequence has exactly 10 of each digit.

2.9. A license plate is 3 letters from the 26 possible repetition allowed followed by 3 digits from 0 thru 9 with repetition allowed. No speeders on the Gwynns Falls Parkways get tickets because the Baltimore speed cameras are weird: they can only record which letters and which digits appeared but not the order they appear on the plate. How many distinct recordings can these camera make?

2.10. Consider the vectors {(x1, x2, x3, x4, x5, x6) : xi ≥ 0, xi integer}.

The following are separate questions:

(a) How many of these vectors have xi = 10?

(b) How many of these vectors have xi = 10 and x1 = x2 = x3 = 1?

(c) How many of these vectors have xi = 10 and x1 = x2 = x3?

h.11. Let 1 ≤ k < n be positive integers. Show that:

You don’t have to be rigorous. This is sometimes called Fermat’s combinatorial identity.

h.12. Let n be a positive integer. Determine the number of n-vectors (x1, x2, . . . , xn) of nonnegative integers such that

Simplify your final answer as much as possible.

h.13. Let n > 0 be a fixed integer. Find the value(s) of 0 ≤ k ≤ n that maximize the value of . You may want to consider the ratios of successive values of k in the binomial coefficients. Also note that k must be an integer.