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Prove expression (1.3.9). Consider Remark 1.3.2. (b) Now prove the general inclusion exclusion formula given by the expression (1.3.13). 1.3.13 Three distinct integers are chosen at random from the first 20 positive integers. Compute the probability that: (a) their sum is even; (b) their product is even.

Suppose D is a nonempty collection of subsets of C. Consider the collection of events Note that φ ∈ B because it is in each σ-field, and, hence, in particular, it is in each σ-field . Continue in this way to show that B is a σ-field.

These assignment focuses on the  effects of using technology in education. Further explain factor that lead to limiting social media to teenagers. Effects of technology in education:Limiting social media to teenagers Choose one of the following topics to write on the causes OR the effects (these are two different things). Some of the topics are

Sexual education policy • Pain management in the time of the opioid epidemic • FDA oversight of medical devices • Marijuana legalization, drug decriminalization, harm reduction policies • Medicare’s lack of coverage for hearing aids, vision, or dental costs • The U.S. government not being able to negotiate drug prices for Medicare beneficiaries • Single-payer

Each of four persons fires one shot at a target. Let denote the event that the target is hit by person compute the probability that (a) all of them hit the target; (b) exactly one hits the target; (c) no one hits the target; (d) at least one hits the target.

A bowl contains three red (R) balls and seven white (W) balls of exactly the same size and shape. Select balls successively at random and with replacement so that the events of white on the first trial, white on the second, and so on, can be assumed to be independent. In four trials, make certain

Cause/Effect Essay Essay directions and topics Choose one of the following topics to write on the causes OR the effects (these are two different things). Some of the topics are specified as only the effects or just the causes. This essay will also give you a chance to incorporate research. If you will notice, these

Let a card be selected from an ordinary deck of playing cards. The outcome c is one of these 52 cards. Let  if c is an ace, let  if c is a king, let  if c is a queen, let  if c is a jack, and let otherwise. Suppose that P assigns a probability of  to each outcome c. Describe the

A chemist wishes to detect an impurity in a certain compound that she is making. There is a test that detects an impurity with probability 0.90; however, this test indicates that an impurity is there when it is not about 5% of the time. The chemist produces compounds with the impurity about 20% of the

For each of the following, find the constant  so that  satisfies the condition of being a pmf of one random variable .