代做553.420/620 Probability Assignment #03代写数据结构程序

553.420/620 Probability

Assignment #03

1. When n and k are nonnegative integers with 0 ≤ k ≤ n, the binomial coefficient can the thought of as the number of subsets of size k from an n-element set and either of the following formulas can be used to compute it:

Now let x ≠ 0 be any real number. With an abuse of notation, when k any nonnegative integer, we can define

For example, and

Let r ≥ 1 and y ≥ 0 both be integers. Show

Remark. The binomial coefficient on the right is called the negative binomial coefficient for this reason.

2. Simplify the following completely with justification. If you are appealing to a theorem or identity, be sure to carefully show what, where, and how it is being used. In all of these expressions n > 0 is an integer.

(a)

(b)

(c) In the expansion (2x − 1)10 what is the coefficient of x5? Justify your assertion.

(d) When n is odd, find a simple expression with justification for

3. Consider a standard deck of 52 cards. The following are separate questions unless noted otherwise.

(a) We deal 13 cards to a player. What’s the probability they have 3 clubs, 3 diamonds, 3 hearts, and 4 spades.

(b) We deal 13 cards to each of 4 players – a Bridge deal. What’s the probability that one player gets all 4 Aces and another player gets all 4 Kings?

4. An urn has 4 balls: 1 blue, 1 green, 1 red and 1 yellow. We draw 4 balls with replacement.

(a) Find the probability we see 2 blue and 2 green balls.

(b) Find the probability we see two of one color and 2 of another.

(c) Find the probability that we are missing at least one color. Please: answer this two ways: one way using inclusion exclusion, another way by considering the complement event.

(d) Find the probability we get exactly two of one color and one each of two other colors.

5. (a) I deal you 5 cards from a deck of 52. How many 5-card hands are possible?

(b) Now I take 5 decks of 52 cards and shuffle them together (for 52 × 5 = 260 cards total). I then deal you 5 cards. How many possible 5-card hands now?

6. KPOT offers twelve (12) different items on their lunch buffet. They have a special that allows guests to select any four (repetition allowed) items from their buffet table. For example, a guest can take all four items to be fried shrimp, for instance. How many selections are possible?

7. A license plate is 3 letters from the 26 possible repetition allowed followed by 3 digits from 0 thru 9 with repetition allowed. The speed cameras on the Gwynns Falls Parkway are weird: they can only record which letters and which digits appeared on the speeding car but not the order they appear on the plate. How many distinct recordings can these cameras make?

8. We have 10 cards numbered 0 thru 9. The cards are shuffled and lined up. If the line up starts or ends with two even digits, we win a prize. Compute the probability we win a prize.

9. How many sequences of coin tosses have exactly y tails before the rth head? You can assume that both y and r are positive integers.

10. (a) In class we illustrated the inclusion-exclusion rules and how one could get the inclusion-exclusion rule for 3 events from that of two events. Your job is to show that the inclusion-exclusion rule 4 sets follows from that of the one for three sets and two sets. Hint: Think of P(A1 ∪ A2 ∪ A3 ∪ A4) as P([A1 ∪ A2 ∪ A3] ∪ A4).

(b) A hat has 12 pieces of paper in it with 4 number 1’s, 4 number 2’s, and 4 number 3’s well-mixed in the hat. A person reaches in and selects 4 pieces of paper at once. What’s the probability they are missing at least one of the numbers in their selection?