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List three methods of assigning probabilities.

If two events are mutually exclusive, can they occur concurrently? Explain.

For each of the following situations, explain why the combinations rule or the permutations rule should be used. (a) Determine the number of different groups of 5 items that can be selected from 12 distinct items. (b) Determine the number of different arrangements of 5 items that can be selected from 12 distinct items.

You need to know the number of different arrangements possible for five distinct letters. You decide to use the permutations rule, but your friend tells you to use \(5 !\). Who is correct? Explain.

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.

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

(a) Draw a tree diagram to display all the possible outcomes that can occur when you flip a coin and then toss a die. (b) How many outcomes contain a head and a number greater than 4 ? (c) Probability extension: Assuming the outcomes displayed in the tree diagram are all equally likely, what is the probability that you will get a head and a number greater than 4 when you flip a coin and toss a die?

(a) Make a tree diagram to show all the possible sequences of answers for three multiple-choice questions, each with four possible responses. (b) Probability extension: Assuming that you are guessing the answers so that all outcomes listed in the tree are equally likely, what is the probability that you will guess the one sequence that contains all three correct answers?

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.)

You roll two fair dice, a green one and a red one. (a) Are the outcomes on the dice independent? (b) Find \(P(1\) on green die and 2 on red die). (c) Find \(P(2\) on green die and 1 on red die). (d) Find \(P((1)\) on green die and 2 on red die) or \((2\) on green die and 1 on red die)).