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Vinodhini John

Vinodhini J.

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A set of 50 numbered discs consists of 8 ones, 12 twos, 14 threes, 7 fours, and 9 fives. If one disc is drawn at random, what is the probability that its number is (i) 2 , (ii) 4 , (iii) 2 or 4 , (iv) $\leq 4,(\mathbf{v})$ odd.

A set of 50 numbered discs consists of 8 ones, 12 twos, 14 threes, 7 fours, and 9 fives. If one disc is drawn at random, what is the probability that its number is (i) 2 , (ii) 4 , (iii) 2 or 4 , (iv) $\leq 4,(\mathbf{v})$ odd.

The Chemistry Maths Book

A single-elimination tournament with four players is to be held. A total of three games will be played. In Game 1 , the players seeded (rated) first and fourth play. In Game 2 , the players seeded second and third play. In Game $3,$ the winners of Games 1 and 2 play, with the winner of Game 3 declared the tournament winner. Suppose that the following probabilities are known: $P($ Seed 1 defeats Seed 4$)=0.8$ $P($ Seed 1 defeats $\operatorname{Seed} 2)=0.6$ $P($ Seed 1 defeats $\operatorname{Seed} 3)=0.7$ $P($ Seed 2 defeats $\operatorname{Seed} 3)=0.6$ $P($ Seed 2 defeats Seed 4$)=0.7$ $P($ Seed 3 defeats Seed 4) $=0.6$ a. How would you use random digits to simulate Game 1 of this tournament? b. How would you use random digits to simulate Game 2 of this tournament? c. How would you use random digits to simulate the third game in the tournament? (This will depend on the outcomes of Games 1 and $2 .$ ) d. Simulate one complete tournament, giving an explanation for each step in the process. e. Simulate 10 tournaments, and use the resulting information to estimate the probability that the first seed wins the tournament. f. Ask four classmates for their simulation results. Along with your own results, this should give you information on 50 simulated tournaments. Use this information to estimate the probability that the first seed wins the tournament. g. Why do the estimated probabilities from Parts (e) and (f) differ? Which do you think is a better estimate of the actual probability? Explain.

Statistics Learning From Data

Probability

Estimating Probabilities Empirically and…
Jay and Maurice are playing a tennis match. In one particular game, they have reached deuce which means each player won three points. Now in order to finish the game, one of two players must get two points ahead of the other. For example, Jay will win if he wins the next two points $(J J),$ or if Maurice wins the next point and Jay the three points after that $(M J J),$ or if the result of the next six points is JMMJJJ, etc. (a) Suppose that the probability of Jay winning a point is.6 and outcomes of successive points are independent of one another. What is the probability that Jay wins the game? [Hint: In the law of total probability, let $A_{1}=\{$ Jay wins each of the next two points $\}, A_{2}=$ \{Maurice wins each of the next two points $\},$ and $A_{3}=\{$ each player wins one of the next two points $\} .$ Also let $p=P($ Jay wins the game). How does $p$ compare to $P($ Jay wins the gamelA $_{3} ) ?$ (b) If Jay wins the game, what is the probability that he needed only two points to do so?

Jay and Maurice are playing a tennis match. In one particular game, they have reached deuce which means each player won three points. Now in order to finish the game, one of two players must get two points ahead of the other. For example, Jay will win if he wins the next two points $(J J),$ or if Maurice wins the next point and Jay the three points after that $(M J J),$ or if the result of the next six points is JMMJJJ, etc. (a) Suppose that the probability of Jay winning a point is.6 and outcomes of successive points are independent of one another. What is the probability that Jay wins the game? [Hint: In the law of total probability, let $A_{1}=\{$ Jay wins each of the next two points $\}, A_{2}=$ \{Maurice wins each of the next two points $\},$ and $A_{3}=\{$ each player wins one of the next two points $\} .$ Also let $p=P($ Jay wins the game). How does $p$ compare to $P($ Jay wins the gamelA $_{3} ) ?$ (b) If Jay wins the game, what is the probability that he needed only two points to do so?

Probability with Applications in Engineering, Science, and Technology

Probability

Supplementary Exercises

Questions asked

ANSWERED

Lucas Finney verified

Numerade educator

In a school machine shop, 60% of all machine breakdowns occur on lathes and 15% occur on drill presses. Let E denote the event that the next machine breakdown is on a lathe, and let F denote the event that a drill press is the next machine to break down. With P(E) = .60 and P(F) = .15, calculate: P(Eᶜ ∩ Fᶜ) 6.91 There are five faculty members in a certain academic department. These individuals have 3, 6, 7, 10, and 14 years of teaching experience. Two of these individuals are randomly selected to serve on a personnel review committee. What is the probability that the chosen representatives have a total of at least 15 years of teaching experience? (Hint: Consider all possible committees.) 6.92 The general addition rule for three events states that P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C) A new magazine publishes columns entitled “Art” (A), “Books” (B), and “Cinema” (C). Suppose that 14% of all subscribers read A, 23% read B, 37% read C, 8% read A and B, 9% read A and C, 13% read B and C, and 5% read all three columns. What is the probability that a randomly selected subscriber reads at least one of these three columns? 6.93 A theater complex is currently showing four R-rated movies, three PG-13 movies, two PG movies, and one G movie. The following table gives the number of people at the first showing of each movie on a certain Saturday: Number of Theater Rating Viewers 1 R 600 2 PG-13 420 3 PG-13 323 4 R 196 5 G 254 6 PG 179 7 PG-13 114 8 R 205 9 R 139 10 PG 87

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ANSWERED

Lucas Finney verified

Numerade educator

Two individuals,A and B, are finalists for a chess championship. They will play a sequence of games, each of which can result in a win for A, a win for B, or a draw. Suppose that the outcomes of successive games are independent, with P(A wins game) ! .3, P(B wins game) ! .2, and P(draw) ! .5. Each time a player wins a game, he earns 1 point and his opponent earns no points. The first player to win 5 points wins the championship. For the sake of simplicity, assume that the championship will end in a draw if both players obtain 5 points at the same time.

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