Homework 3: Probability And Statistics

University:
California State University, Northridge
Course:
MATH 340 | Introduction to Probability and Statistics
Academic year:

2023
Views:

403

Pages:

5
Author:

Nyi Moe

11/19/23, 11:36 PM HW03-Nyi Nyi Moe Htet Student: Nyi Nyi Moe Htet Date: 11/19/23 Instructor: Xiaolong Han Course: CSUN_MATH340_HAN (1) Assignment: HW03 1. Let A ⊂ B. Express the following probabilities as simply as possible. P(A∣B). Choose the correct answer below. A. 1 B. 0 D. P(B) G. P A c c P(B) P(AB) I. c L. P(AB) P(B) P(B) P(A) . Choose the correct answer below. P AB c B. F. P(B) C. P B P(B) P(A) G. P AB J. M. P(AB) c D. 1 P(A) P(A) I. P(A) P(A) K. P B P(B) E. F. c c P(AB) A. P(AB) H. P A B P(A) J. P A∣ B E. C. P B c P(AB) P(A) H. P(A) K. P A c L. c N. 0 O. P AB c P. P B P B c c P(A) P(A) P(B) P(B∣A). Choose the correct answer below. c c B. P(A) A. P A B P(B) P(A) P(B) P(AB) P(B) H. P A c I. K. P B c L. P(AB) 1 P(A) P(B) J. c . Choose the correct answer below. c A. F. P(A) G. P B∣ A E. 0 P(AB) D. C. P A B B. P(B) E. 1 F. P(B) P(A) P(A) C. c P(A) D. P A B P(B) G. 0 P(B) H. I. P(AB) P(B) M. P(AB) J. P A c K. P(B) N. P(B) L. P A c c P(A) c O. https://tdx.acs.pearsonprd.tech/api/v1/print/highered P(AB) P A P A B P A P. P B c c 1/5 11/19/23, 11:36 PM HW03-Nyi Nyi Moe Htet 2. Complete parts (a) and(b). (a) Prove that if E and F are mutuallyexclusive, then the following equation is true. P(E∣E ∪ F) = P(E) P(E) + P(F) Based on the definition of conditional probability, P(E∣E ∪ F) = Based directly on an axiom of probability, P(E{E ∪ F}) P(E ∪ F) . if E and F are mutuallyexclusive, P(E ∪ F) = P(E) + P(F). Logically, P(E{E ∪ F}) = P(E). Therefore, the given equation is true. (b) Prove that if Ei , i ≥ 1 are mutuallyexclusive, then the following equation is true. ∞ P Ej ∣ ∪ Ei = i=1 P Ej ∞ ∑ P Ei i=1 ∞ Based on the definition of conditional probability, P Ej ∣ ∪ ∞ Ei = P Ej i=1 an axiom of probability, P ∪ ∪ ∞ Ei i=1 divided by P ∪ Ei . i=1 ∞ ∞ By ∩ Ei i=1 = ∑ P Ei . Logically, since Ei , i ≥ 1 are mutually exclusive and Ej = Ei i=1 ∞ for exactly onei, P Ej ∩ ∪ Ei i=1 = P Ej . Therefore, the given equation is true. https://tdx.acs.pearsonprd.tech/api/v1/print/highered 2/5 11/19/23, 11:36 PM HW03-Nyi Nyi Moe Htet 3. Consider two independent tosses of a fair coin. Let A be the event that the first toss results inheads, let B be the event that the second toss results inheads, and let C be the event that in both tosses the coin lands on the same side. Show that the eventsA, B, and C are pairwise independent—that is, A and B areindependent, A and C areindependent, and B and C are independent—but not independent. Which of the following must be true of any eventsA, B, and C if and only if they are pairwiseindependent, but is not sufficient to conclude that the three events areindependent? A. P(AB) = P(A∣B), P(BC) = P(B∣C), and P(AC) = P(A∣C) B. P(AB) = P(BC), P(BC) = P(AC), and P(AC) = P(AB) C. P(A∣B) = P(B∣C), P(B∣C) = P(C∣A), and P(C∣A) = P(A∣B) D. P(AB) = P(A)P(B), P(BC) = P(B)P(C), and P(AC) = P(A)P(C) E. P(A∣B) = P(B), P(B∣C) = P(C), and P(C∣A) = P(A) In the order they appear in the previousstep, the six expressions across the three equations in the previous step are equal to thefollowing: 1 4 , 1 4 , 1 4 1 , 4 , 1 , and 4 1 4 . Therefore, the eventsA, B, and C are pairwise independent. (Simplify youranswers.) Which of the following must be true of any eventsA, B, and C if and only if they areindependent? A. P(ABC) = P(A)P(B∣A)P(C∣B) B. P(A)P(B)P(C) = P(AB)P(BC)P(AC) C. P(ABC) = P(A)P(B)P(C) D. P(A)P(B∣A)P(C∣AB) = P(ABC) The left side of the equation in the previous step is equal to 1 4 and the right side of the equation in the previous step is equal 1 to . Since these are notequal, eventsA, B, and C are not independent. 8 (Simplify youranswers.) 4. In each of n independent tosses of acoin, the coin lands on heads with probability p. How large need n be so that the probability of 1 obtaining at least one head is at least ? 2 Write an expression for the probability that there is at least one head in n tosses of the coin. 1 − (1 − p) n To find how large n must be so that the probability of obtaining at least one head is at least 1 2 , solve an appropriate inequality using the expression for the probability that there is at least one head in n tosses of the coin. n≥ − log 2 log (1 − p) https://tdx.acs.pearsonprd.tech/api/v1/print/highered 3/5 11/19/23, 11:36 PM HW03-Nyi Nyi Moe Htet 5. The probability of getting a head on a single toss of a coin is p. Suppose that A starts and continues to flip the coin until a tail showsup, at which point B starts flipping. Then B continues to flip until a tail comesup, at which point A takesover, and so on. Let Pn,m denote the probability that A accumulates a total of n heads before B accumulates m. Show that the following equation is true. Pn,m = pPn − 1,m + (1 − p) 1 − Pm,n Let H1 denote the event thatA's first flip is heads and let E denote the event that A accumulates a total of n heads before B accumulates m. The most useful first step is to apply the law of total probability to note that Pn,m is equal to P H1 P E∣H1 c plus c P H 1 P E∣ H 1 . The probability thatA's first flip is heads is p . IfA's first flip isheads, then the second overall flip is made by A. In order for the player making the second overall flip to accumulate their designated total of headsfirst, the player making the second overall flip must accumulate n−1 more heads afterA's first flip before the other player accumulates m more heads afterA's first flip. (Simplify youranswers.) The probability thatA's first flip is not heads is 1−p . IfA's first flip isheads, then the second overall flip is made by B. In order for the player making the second overall flip to accumulate their designated total of headsfirst, the player making the second overall flip must accumulate m more heads afterA's first flip before the other player accumulates n more heads afterA's first flip. (Simplify youranswers.) If x is the probability that B accumulates their designated total of headsfirst, then 1−x their designated total of heads first. Combining all of thisinformation, the given equation is true. (Simplify youranswer.) https://tdx.acs.pearsonprd.tech/api/v1/print/highered is the probability that A accumulates 4/5 11/19/23, 11:36 PM HW03-Nyi Nyi Moe Htet 6. Independent trials that result in a success with probability p are successively performed until a total of r successes is obtained. Show that the probability that exactly n trials are required is given by the following expression. n−1 r−1 r p (1 − p) n−r Use this result to solve the problem of thepoints, which seeks to identify the probability that r successes occur before m failures. Hint: In order for it to take n trials to obtain rsuccesses, how many successes must occur in the first n − 1 trials? In order for it to take n trials to obtain rsuccesses, the number of successes that must occur in the first n − 1 trials is the number of failures that must occur in the first n − 1 trials is occurs in the first n − 1 trials. (Simplify youranswers.) n−r r−1 and . Let B be the event that this many successes and failures Consider any particular sequence of the first n − 1 outcomes containing the numbers of successes and failures identified in the previous step. Each one of these sequenceswill, by the assumed independence oftrials, occur with probability p The number of such sequences is equal to the number of combinations of (Simplify youranswers.) r−1 r−1 (1 − p) items selected from n−r . n−1 items. The probability that it takes exactly n trials to obtain r successes is equal to the probability of each sequence times the number of combinations of such sequences. required to obtain a total of r successes is equal to pP(B). The probability that exactly n trials are So, the given expression is the probability that exactly n trials are required. Now use the expression to solve the problem of thepoints, which seeks to identify the probability that r successes occur before m failures. In terms of r andm, the minimum number of trials that could be run for r successes to occur before m failures is r , and the maximum number of trials that could be run for r successes to occur before m failures is m+r−1 . (Simplify youranswers.) So, the probability that r successes occur before m failures is m+r−1 n−1 r ∑ p (1 − p) r−1 n= r (Simplify youranswers.) n−r . 7. Prove directly that P(E∣F) = P(E∣FG)P(G∣F) + P E∣FGc P Gc ∣F . The probability P(E∣F) can be rewritten as P(E∣F) = P E∣FG c c P G ∣F = P EFG P(F) P(EF) P(F) . Similarly, P(E∣FG)P(G∣F) = P(EFG) P(F) c . The given statement is true since P(EF) = P(EFG) + P EFG c and . 8. This question has been removed from this assignment by your instructor; you have received full credit. 9. This question has been removed from this assignment by your instructor; you have received full credit. 10. This question has been removed from this assignment by your instructor; you have received full credit. https://tdx.acs.pearsonprd.tech/api/v1/print/highered 5/5