probability

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Mathematics

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this is a probability hw

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MATH 331 – Homework 2 Instructor: Dr. Brown 1. We roll a fair die repeatedly until we see the number four appear and then we stop. a. Following an example discussed in class, describe a sample space Ω and a probability measure P to model this situation. b. Calculate the probability that the number four never appears. c. Let A be the event that at most 9 rolls are needed to see the first four. Find the probability Pr(A). (Check Equation that shows how to evaluate a partial sum of geometric type.) 2. We break a stick at a uniformly chosen random location. Find the probability that the shorter piece is less than 1/5th of the original. 3. Pick a uniformly chosen random point inside a unit square and draw a circle of radius 1/3 around the point. Find the probability that the circle lies entirely inside the square. 4. a. Let (X, Y ) denote a uniformly chosen random point inside the unit square [0, 1]2 = [0, 1] × [0, 1] = {(x, y) : 0 ≤ x, y ≤ 1}. Let 0 ≤ a < b ≤ 1. Find the probability Pr(a < X < b), that is, the probability that the x-coordinate X of the chosen point lies in the interval (a, b) b. What is the probability Pr(|X − Y | ≤ 1/4)? 5. We roll a fair die repeatedly until we see the number four appear and then we stop. What is the probability that we needed an even number of die rolls? 6. Sixty percent of the students at a certain school wear neither a watch nor a bracelet. Twenty five percent wear a watch and thirty percent wear a bracelet. a. If one of the students is chosen at random what is the probability that this student is wearing a watch or a bracelet? b. What is the probability that this student is wearing a watch and a bracelet? 7. An urn contains four balls: 1 white, 1 green and 2 red. I draw 3 balls with replacement. Find the probability that I did not see all three colors through two different calculations, as specified by (a) and (b) below. a. Define the event W = { white ball appeared } and similarly for G and R. Use inclusion-exclusion. b. Use the complement. 8. Assume that Pr(A) = 0.4 and Pr(B) = 0.7. Making no further assumptions on A and B, show that Pr(AB) satisfies 0.1 ≤ Pr(AB) ≤ 0.4. 9. Show that for any events A1 , A2 , ..., An , Pr(A1 ∪ A2 ∪ ... ∪ An ) ≤ Pn k=1 Pr(Ak )
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Running head: MATH 331

Math 331
Homework 2
Instructor: Dr. Brown

MATH 331

2

1. We roll a fair die repeatedly until we see the number four appear and then we stop.
a) Following the example discussed in class, describe a sample space Ω and a
probability measure P to model this situation.
Solution

Modelling the outcomes of the experiment as the number of times we rolled the
die we take Ω={∞, 1,2,3,…}.
For k≥ 1, we have:

P(k)=P{needed k rolls}=P{no fours in the first k-1 rolls, then a 4}.
Counting the number of ways we cannot roll a four on each roll (5), we have
5
1
P(k)=P{no fours in the first k-1 rolls, then a 4}= ( )𝑘�...


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