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What is the law of large numbers? If you were using the relative frequency of an event to estimate the probability of the event, would it be better to use 100 trials or 500 trials? Explain.

(a) Draw a tree diagram to display all the possible head-tail sequences that can occur when you flip a coin three times. (b) How many sequences contain exactly two heads? (c) Probability Extension Assuming the sequences are all equally likely, what is the probability that you will get exactly two heads when you toss a coin three times?

Basic Computation: Rules of Probability Given \(P\left(A^{c}\right)=0.8, P(B)=0.3\) \(P(B | A)=0.2:\) (a) Compute \(P(A \text { and } B).\) (b) Compute \(P(A \text { or } B).\)

Counting Four wires (red, green, blue, and yellow) need to be attached to a circuit board. A robotic device will attach the wires. The wires can be attached in any order, and the production manager wishes to determine which order would be fastest for the robot to use. Use the multiplication rule of counting to determine the number of possible sequences of assembly that must be tested. Hint: There are four choices for the first wire, three for the second, two for the third, and only one for the fourth.

Greg made up another question for a small quiz. He assigns the probabilities \(P(A)=0.6, P(B)=0.7, P(A | B)=0.1\) and asks for the probability \(P(A \text { or } B\) ). What is wrong with the probability assignments?

Consider a family with 3 children. Assume the probability that one child is a boy is 0.5 and the probability that one child is a girl is also \(0.5,\) and that the events "boy" and "girl" are independent. (a) List the equally likely events for the gender of the 3 children, from oldest to youngest. (b) What is the probability that all 3 children are male? Notice that the complement of the event "all three children are male" is "at least one of the children is female." Use this information to compute the probability that at least one child is female.

Critical Thinking Suppose two events \(A\) and \(B\) are mutually exclusive, with \(P(A) \neq 0\) and \(P(B) \neq 0 .\) By working through the following steps, you'll see why two mutually exclusive events are not independent. (a) For mutually exclusive events, can event \(A\) occur if event \(B\) has occurred? What is the value of \(P(A | B) ?\) (b) Using the information from part (a), can you conclude that events \(A\) and \(B\) are not independent if they are mutually exclusive? Explain.

Consider the experiment of tossing a fair coin 3 times. For each coin, the possible outcomes are heads or tails. (a) List the equally likely events of the sample space for the three tosses. (b) What is the probability that all three coins come up heads? Notice that the complement of the event "3 heads" is "at least one tail." Use this information to compute the probability that there will be at least one tail.

You toss a pair of dice. (a) Determine the number of possible pairs of outcomes. (Recall that there are six possible outcomes for each die.) (b) There are three even numbers on each die. How many outcomes are possible with even numbers appearing on each die? (c) Probability extension: What is the probability that both dice will show an even number?

Critical Thinking Consider the following events for a driver selected at random from the general population: \(A=\) driver is under 25 years old \(B=\) driver has received a speeding ticket Translate each of the following phrases into symbols. (a) The probability the driver has received a speeding ticket and is under 25 years old (b) The probability a driver who is under 25 years old has received a speeding ticket (c) The probability a driver who has received a speeding ticket is 25 years old or older (d) The probability the driver is under 25 years old or has received a speeding ticket (e) The probability the driver has not received a speeding ticket or is under 25 years old