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To win the 3 -of- 10 lottery game, players must match three numbers from 1 to 10 with those selected in a random drawing. Remember that order doesn't matter. a. How many possible triples are there in the 3 of \(-10\) lottery game? b. What are your chances of winning the 3 of \(-10\) lottery game?

The 鈥淪huffle鈥� button on Tamika鈥檚 CD player plays the songs in a random order. Tamika puts a four-song CD into the player and presses 鈥淪huffle.鈥� a. How many ways can the four songs be ordered? b. What is the probability that Song 1 will be played first? c. What is the probability that Song 1 will not be played first? d. Songs 2 and 3 are Tamika鈥檚 favorites. What is the probability that one of these two songs will be played first? e. What is the probability that Songs 2 and 3 will be the first two songs played (in either order)?

Suppose you roll two 12 -sided dice with faces numbered 1 to \(12 .\) a. How many possible number pairs can you roll? b. What is the greatest sum possible from a roll of two 12 -sided dice? c. What sum is most likely? What is the probability of this sum?

Design a championship structure for six teams in which all teams play the first round. Assuming each team is equally likely to win a single game, find the probability that each team will win the championship.

Suppose the \(3-o f-7\) lottery game was modified so that after each number was selected, that number was placed back into the group before the next number was selected. In this way, a number could be repeated, meaning triples such as \(1-2-2\) and \(3-3-3\) would be possible. a. How many possible pairs are there for this modified game, assuming that order does matter? Explain. b. Since order really doesn't matter in this game, \(1-1-2,1-2-1,\) and \(2-1-1\) are all the same triple. So there are only 84 possible different triples. Are all of these different triples equally likely? Explain. c. If you choose one number triple for this modified game, what is the probability you will win. (Hint: There are three cases to consider.)

Imagine rolling five regular dice and looking for outcomes when all five dice match. a. How many different outcomes are possible on a roll of five dice? Explain. b. In how many of the possible outcomes do all five dice match? c. What is the probability of getting all five dice to match on a single roll? d. Suppose Tamika is given three rolls to get five matching dice. On the second and third rolls, she may roll some or all of the five dice again. On her first roll, Tamika gets three \(3 \mathrm{s}, \mathrm{a} 2\) and a \(6 .\) She picks up the dice showing 2 and 6 and rolls them again. What is the probability that she will get two more 3 \(\mathrm{s}\) on this roll?

Challenge In Investigation \(2,\) you always assumed that two teams have the same chances of winning a single game. For this exercise, assume that Team \(\mathrm{A}\) has a 60\(\%\) chance of defeating Team \(\mathrm{B}\) in every game they play against each other. a. Suppose there is a one-game tournament between the teams and the winner of the game wins the tournament. What is the probability that Team \(\mathrm{A}\) will win? That Team \(\mathrm{B}\) will win? b. Use a tree diagram to show all the possibilities for the tournament. For example, in the first game, there are two branches: A wins or \(\mathrm{B}\) wins. (Hint: If \(\mathrm{A}\) wins the first two games, is a third game played?) c. Suppose the teams played \(1,000\) tournaments. In how many tournaments would you expect Team \(\mathrm{A}\) to win the first game? In how many of those tournaments would you expect Team \(\mathrm{A}\) to also win the second game? d. For each combination in your tree diagram, use similar reasoning to find the number of tournaments out of \(1,000\) you would expect to go that way. For example, one combination should be ABB; in how many tournaments out of \(1,000\) would you expect the winner to be \(\mathrm{A},\) then \(\mathrm{B}\) , and then \(\mathrm{B}\) ? (Hint. Check your answers by adding them; they should total to \(1,000 . )\) e. Find the total number of tournaments out of \(1,000\) in which each team wins the tournament. What is the probability that Team \(\mathrm{A}\) wins a tournament? f. Which tournament, one-game or best-two-out-of-three, is better for Team B?

Consider the line \(y=-5 x-7\) a. A second line is parallel to this line. What do you know about the equation of the second line? b. Write an equation for the line parallel to \(y=-5 x-7\) that passes through the origin. 6\. Write an equation for the line parallel to \(y=-5 x-7\) that crosses the \(y\) -axis at the point \((0,-2) .\) d. Write an equation for the line parallel to \(y=-5 x-7\) that passes through the point \((3,0) .\)

Expand and simplify each expression. $$ -3 a(2 a-3) $$