Finding Probabilities In Exercises 51-60, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. $$ \begin{array}{lll} ext { 51. } P(z-1.85) & ext { 55. } P(-0.892.58) & ext { 60. } P(z1.54)\end{array} $$

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Finding Probabilities In Exercises 51-60, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. $$ \begin{array}{lll} ext { 51. } P(z<1.45) & ext { 52. } P(z<-0.18) & ext { 53. } P(z>2.175) \ ext { 54. } P(z>-1.85) & ext { 55. } P(-0.892.58) & ext { 60. } P(z<-1.54 ext { or } z>1.54)\end{array} $$

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Alright, I need to find the probabilities for exercises 51 through 60 using the standard normal distribution. Let's tackle them one by one. **51. $$ P(z < 1.45) $$** First, I recall that the standard normal distribution is symmetric around zero, and the cumulative distribution function (CDF) gives the probability that a standard normal random variable is less than a given value. Using a standard normal table or a calculator, I can find that $$ P(z < 1.45) $$ is approximately 0.9265. **52. $$ P(z < -0.18) $$** Since the standard normal distribution is symmetric, $$ P(z < -0.18) $$ is the same as $$ 1 - P(z < 0.18) $$. From the table, $$ P(z < 0.18) $$ is about 0.5714, so $$ P(z < -0.18) $$ is $$ 1 - 0.5714 = 0.4286 $$. **53. $$ P(z > 2.175) $$** This is the probability that $$ z $$ is greater than 2.175. Using the standard normal table, $$ P(z < 2.175) $$ is approximately 0.9850. Therefore, $$ P(z > 2.175) = 1 - 0.9850 = 0.0150 $$. **54. $$ P(z > -1.85) $$** This is the probability that $$ z $$ is greater than -1.85. Using the symmetry of the standard normal distribution, $$ P(z > -1.85) = 1 - P(z < -1.85) $$. From the table, $$ P(z < -1.85) $$ is about 0.0322, so $$ P(z > -1.85) = 1 - 0.0322 = 0.9678 $$. **55. $$ P(-0.89 < z < 0) $$** This is the probability that $$ z $$ is between -0.89 and 0. Using the standard normal table, $$ P(z < 0) = 0.5 $$ and $$ P(z < -0.89) $$ is approximately 0.1867. Therefore, $$ P(-0.89 < z < 0) = P(z < 0) - P(z < -0.89) = 0.5 - 0.1867 = 0.3133 $$. **56. $$ P(0 < z < 0.525) $$** This is the probability that $$ z $$ is between 0 and 0.525. From the table, $$ P(z < 0.525) $$ is approximately 0.7000, and $$ P(z < 0) = 0.5 $$. Therefore, $$ P(0 < z < 0.525) = P(z < 0.525) - P(z < 0) = 0.7000 - 0.5 = 0.2000 $$. **57. $$ P(-1.65 < z < 1.65) $$** This is the probability that $$ z $$ is between -1.65 and 1.65. Using the standard normal table, $$ P(z < 1.65) $$ is approximately 0.9505 and $$ P(z < -1.65) $$ is about 0.0505. Therefore, $$ P(-1.65 < z < 1.65) = P(z < 1.65) - P(z < -1.65) = 0.9505 - 0.0505 = 0.9000 $$. **58. $$ P(-1.54 < z < 1.54) $$** Similarly, $$ P(z < 1.54) $$ is approximately 0.9382 and $$ P(z < -1.54) $$ is about 0.0618. Therefore, $$ P(-1.54 < z < 1.54) = 0.9382 - 0.0618 = 0.8764 $$. **59. $$ P(z < -2.58 ext{ or } z > 2.58) $$** This is the probability that $$ z $$ is less than -2.58 or greater than 2.58. Using the standard normal table, $$ P(z < -2.58) $$ is approximately 0.0049 and $$ P(z > 2.58) = 1 - P(z < 2.58) \approx 1 - 0.9949 = 0.0051 $$. Therefore, $$ P(z < -2.58 ext{ or } z > 2.58) = P(z < -2.58) + P(z > 2.58) = 0.0049 + 0.0051 = 0.0100 $$. **60. $$ P(z < -1.54 ext{ or } z > 1.54) $$** Similarly, $$ P(z < -1.54) $$ is approximately 0.0618 and $$ P(z > 1.54) = 1 - P(z < 1.54) \approx 1 - 0.9382 = 0.0618 $$. Therefore, $$ P(z < -1.54 ext{ or } z > 1.54) = 0.0618 + 0.0618 = 0.1236 $$. 51. $$ P(z < 1.45) = 0.9265 $$ 52. $$ P(z < -0.18) = 0.4286 $$ 53. $$ P(z > 2.175) = 0.0150 $$ 54. $$ P(z > -1.85) = 0.9678 $$ 55. $$ P(-0.89 < z < 0) = 0.3133 $$ 56. $$ P(0 < z < 0.525) = 0.2000 $$ 57. $$ P(-1.65 < z < 1.65) = 0.9000 $$ 58. $$ P(-1.54 < z < 1.54) = 0.8764 $$ 59. $$ P(z < -2.58 ext{ or } z > 2.58) = 0.0100 $$ 60. $$ P(z < -1.54 ext{ or } z > 1.54) = 0.1236 $$

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