C1. Let A and B be two events with P(A) = 0.8 and P(B) = 0.4. The following questions concern thevalue of P(An B), the probability that both A and B occur.(a) Prove the upper bound P(AN B) < 0.4.(b) Prove that this upper bound can be achieved, by giving an example of a sample space N, aprobability measure P and events A, B CN such that P(A) = 0.8 and P(B) = 0.4, with equalityP(AN B) = 0.4 in the upper bound.(c) Give a lower bound for P(AN B) – that is, prove that P(A n B) > something – and show thatthis lower bound can be achieved. (Try to work out the correct bound, even if you can't formallyprove it.)C2. A “random digit" is a number chosen at random from {0,1, ...,9}, each with equal probability. Astatistician choses n random digits.(a) For k = 0, 1,... , 9, let A be the event that all the digits are k or smaller. What is the probabilityof A, as a function of k and n?(b) Let Br be the event that the largest digit chosen is equal to k. What is the probability of B?Justify your answer carefully.

We have to use basic probability theory to find out the answers.

Answered: C1. Let A and B be two events with P(A)…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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9.) At the Wimbledon Tennis Championship, to win a match in men’s singles a player must win the best of five sets. The following data represent the number of sets played, X, in the men’s singles final match for the years 1968 to 2011. X 3 4 5 Frequency 19 12 13 (a) Construct a probability model for the random variable, X, the number of sets played in the Wimbledon men’s singles final match.(b) Draw a probability histogram.(c) Compute the mean and standard deviation of the random variable X.
3.) In an experiment on the behavior of young children, each subject is placed in an area with 5 toys. Past experiments have shown that the probability distribution of the number X of toys played with by a randomly selected subject is as follows. Number of Toys Probability 2 3 4 5 0.03 0.16 0.30 0.23 0.17 a.) Find the P(X = 5). b.) If a subject is chosen at random, what is the probability they played with more than 2 toys? c.) If a subject is chosen at random, what is the probability they played with less than 3 toys? d.) Find the expected number of toys played with. e.) Find the standard deviation of the number of toys played with.
Let A and B be two events in a sample space S such that P(A) = 0.5, P(B) = 0.5, and P(A ∩ B) = 0.24. Find the probabilities below. (a) P(A|B)(b) P(B|A)
An experiment can result in one of five equally likely simple events, E,, E, E. Events A, B, and C are defined as follows. A: E,, E, P(A) = 0.4 B: E,, E E E, P(B) = 0.8 C: E,. E, P(C) = 0.4 Refer to the following probabilities. P(AN B) = 0.2 P(A|B) = 0.25 P(B|A) = 0.5 P(B U C) = 1 P(B|C) = 0.5 P(C|B) = 0.25 Use the Addition and Multiplication Rules to find the following probabilities. (a) P(A U B) P(A U B) = --Select-- v+ --Select-- ---Select-- v %3D (b) P(ANB) P(ANB) =-Select-- v P(B) = (c) P(B nC) P(BnC) = -Select-- V- P(C) = Do the results agree with those obtained by listing the simple events in each? P(A U B) = P({E, E, E,, E, E;}) = 1 P(A N B) = P({E,}) = 0.2 P(Bn C) = P({E,}) = 0.2 O Yes O No

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