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A Normal Approximation Probabilities are all around us: . The probability (or likelihood) of winning the Powerball (assuming you bought a ticket) is one in 292 million . The odds of getting attacked by a shark is one in 3.7 million]. . The likelihood that you will die in an airplane crash is one in 205.6 thousand. . The probability of being born with 11 fingers or toes is one in 500. Statistics are bantered about quite often by news reporters, sports broadcasters, politicians, scientists, etc. But, how do people arrive at these values? It may not come as a surprise that the answer is calculus. Whenever one is investigating random events (i.e., events with a likelihood or probability of happening but which are not determined or guaranteed) then one must first determine an underlying probability density function (PDF) that models the random event. We will study PDFs in greater detail later in the semester; however, today we will investigate one of the most common and useful distributions, the standard normal distribution function: 4 ( 20 ) = - 1 - 2 2 / 2 V27 1. The function above provides the "likelihood" of an event happening. For example, under certain conditions, the probability that a particle is a nano-meters to the left or right of its original position (left if x 0) after one second is given by 4(I). (a) What is the likelihood that the particle hasn't moved after one second? (Round your answer to two decimal places) (b) What is the likelihood that the particle is 1 nano-meter to the right of where it started? (Round your answer to two decimal places) In practice it is more common to investigate the likelihood that a random event occurred within a range. For example, in the above scenario it would be more common to investigate how likely it would be for the the particle to be found within 1 nano-meter from where it started after one second. To answer such a question, one must "add up" the probabilities that the particle could be r units from its start for -1