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Calculator Components Problem: The "heart", of a calculator is one or more chips on which thousands of components are etched. Chips are mass produced and have a fairly high probability of being defective. Suppose that a particular brand of calculator uses two kinds of chip. Chip \(A\) has a probability of \(70 \%\) of being defective, and chip \(B\) has a probability of \(80 \%\) of being defective. If one chip of each kind is randomly selected, find the probability that a. Both chips are defective b. \(A\) is not defective c. \(B\) is not defective d. Neither chip is defective e. At least one chip is defective

A card is drawn at random from a standard 52 -card deck. a. What term is used in probability for the act of drawing the card? b. How many outcomes are in the sample space? c. How many outcomes are in the event "the card is a face card"? d. Calculate \(P\) (the card is a face card). e. Calculate \(P(\text { the card is black })\) f. Calculate \(P\) (the card is an ace). g. Calculate \(P\) (the card is between 3 and 7 inclusive). h. Calculate \(P(\) the card is the ace of clubs). i. Calculate \(P\) (the card belongs to the deck). j. Calculate \(P\) (the card is a joker).

Uranium Fission Problem: When a uranium atom splits ("fissions"), it releases 0,1,2,3 or 4 neutrons. Let \(P(x)\) be the probability that \(x\) neutrons are released. Assume the probability distribution is $$\begin{array}{ll}x & P(x) \\\\\hline 0 & 0.05 \\\1 & 0.2 \\\2 & 0.25 \\\3 & 0.4 \\\4 & 0.1\end{array}$$ a. What is the mathematically expected number of neutrons released per fission? b. The number of neutrons released in any one fission must be an integer. How do you explain the fact that the mathematically expected number in part a is not an integer?

The Hawaiian alphabet has 12 letters. How many permutations could be made using a. Two different letters b. Four different letters c. Twelve different letters

Archery Problem 3: An expert archer has the probabilities of hitting various rings shown on the target (Figure \(9-8 a\) ). $$\begin{array}{lcc}\text { Color } & \text { Probability } & \text { Points } \\\\\hline \text { Gold } & 0.20 & 9 \\\\\text { Red } & 0.36 & 7 \\\\\text { Blue } & 0.23 & 5 \\\\\text { Black } & 0.14 & 3 \\\\\text { White } & 0.07 & 1\end{array}$$ a. What is her mathematically expected number of points on any one shot? b. If she shoots 48 arrows, what would her expected score be?

Multiple-Choice Test Problem 1: A short multiple-choice test has four questions. Each question has five choices, exactly one of which is right. Willie Passitt has not studied for the test, so he guesses answers at random a. What is the probability that his answer on a particular question is right? What is the probability that it is wrong? b. Calculate his probabilities of guessing \(0,1,2,3,\) and 4 answers right. c. Perform a calculation that shows that your answers to part b are reasonable. d. Plot the graph of this probability distribution. e. Willie will pass the test if he gets at least three of the four questions right. What is his probability of passing? f. This binomial probability distribution is an example of a function of \(-?\)

Fran Tick takes a 10 -problem precalculus test. The problems may be worked in any order. a. In how many different orders could she work all 10 of the problems? b. In how many different orders could she work any 7 of the 10 problems?

Evaluate the number of combinations or permutations two ways: a. Using factorials, as in the examples of this section b. Directly, using the built-in features of your grapher $$27 C_{19}$$

Traffic Light Problem 1: Two traffic lights on Broadway operate independently. Your probability of being stopped at the first light is \(40 \% .\) Your probability of being stopped at the second one is \(70 \% .\) Find the probability of being stopped at a. Both lights b. Neither light c. The first light but not the second d. The second light but not the first e. Exactly one of the lights

Seed Germination Problem: A package of seeds for an exotic tropical plant states that the probability that any one seed germinates is \(80 \% .\) Suppose you plant four of the seeds. a. Find the probabilities that exactly 0,1,2,3 and 4 of the seeds germinate. b. Find the mathematically expected number of seeds that will germinate.