Normal Distribution: P(z <= A) = 0.7116, Find P(z >= A)

When dealing with a standard normal distribution, understanding the relationship between probabilities is key to solving problems like this. You've been given that $P(z \leq a) = 0.7116$, and you need to find the value of $P(z \geq a)$. This might seem a little tricky at first glance, but it relies on a fundamental property of probability distributions: the total probability under the curve is always 1. Let's break down what this means and how we can use it to find our answer. The standard normal distribution, often denoted by the variable $z$, is a bell-shaped curve that is symmetric around its mean, which is 0. The total area under this curve represents the total probability, which is always equal to 1. The area to the left of a certain value $a$ represents the probability that a random variable from this distribution will be less than or equal to $a$, denoted as $P(z \leq a)$. Conversely, the area to the right of $a$ represents the probability that the random variable will be greater than or equal to $a$, denoted as $P(z \geq a)$. Since the entire area under the curve is 1, the sum of the area to the left of $a$ and the area to the right of $a$ must equal 1. Mathematically, this can be expressed as: $P(z \leq a) + P(z \geq a) = 1$. This equation is the cornerstone of solving our problem. We are given $P(z \leq a) = 0.7116$, and we need to find $P(z \geq a)$. By rearranging the formula, we get $P(z \geq a) = 1 - P(z \leq a)$. Now, all we need to do is substitute the given value into this equation. So, $P(z \geq a) = 1 - 0.7116$. Performing this simple subtraction, we find that $P(z \geq a) = 0.2884$. This means that the probability of a randomly selected value from this standard normal distribution being greater than or equal to $a$ is 0.2884.

Understanding the Standard Normal Distribution and Its Properties

The standard normal distribution is a cornerstone of statistics, providing a standardized way to analyze data that follows a bell curve. Its unique properties make it incredibly useful for making predictions and drawing conclusions about populations. At its heart, the standard normal distribution is characterized by a mean of 0 and a standard deviation of 1. This standardization allows us to compare data from different sources, even if they have different means and standard deviations, by converting them into z-scores. A z-score essentially tells us how many standard deviations a particular data point is away from the mean. The distribution's shape is symmetrical, meaning that the left half of the curve is a mirror image of the right half. This symmetry is crucial because it implies that the probability of a value being a certain distance below the mean is the same as the probability of a value being the same distance above the mean. For instance, the probability of $z \leq -1$ is equal to the probability of $z \geq 1$. The total area under the curve of a standard normal distribution is always 1. This represents the certainty that any value drawn from the distribution will fall somewhere within its range. This total probability of 1 is divided into areas to the left and right of any given point, $a$. The area to the left of $a$, $P(z \leq a)$, represents the cumulative probability up to that point. The area to the right of $a$, $P(z \geq a)$, represents the probability of exceeding that point. Because these two areas together make up the entire distribution, their sum must always equal 1: $P(z \leq a) + P(z \geq a) = 1$. This fundamental relationship is what allows us to solve problems where one probability is known and the other needs to be found. If you know the probability of a value being less than or equal to $a$, you can easily find the probability of it being greater than or equal to $a$ by subtracting the known probability from 1. This principle is not just theoretical; it's applied in various fields, from quality control in manufacturing to risk assessment in finance, helping professionals make informed decisions based on statistical likelihoods. The visual representation of this is our familiar bell curve, where the area under specific segments corresponds to probabilities, and the sum of all these segments is the total probability of 1.

Solving for P(z >= a) Using Complementary Probability

In probability theory, the concept of complementary probability is a powerful tool, especially when dealing with continuous distributions like the standard normal distribution. We've established that the total probability for any event must sum to 1. For a standard normal distribution, this means that the sum of the probability of an event occurring and the probability of that same event not occurring is always 1. In our specific problem, the event is related to the value $a$. We are given $P(z \leq a) = 0.7116$. This value, $0.7116$, represents the probability that a randomly chosen value from the standard normal distribution will be less than or equal to $a$. This is the area under the normal curve to the left of $a$. The question asks for $P(z \geq a)$, which is the probability that a randomly chosen value will be greater than or equal to $a$. This corresponds to the area under the normal curve to the right of $a$. Since the entire area under the curve represents a total probability of 1, the area to the left of $a$ ($P(z \leq a)$) and the area to the right of $a$ ($P(z \geq a)$) must add up to 1. This is where the complementary probability comes into play. The event "$z \geq a$" is the complement of the event "$z < a