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Question: This problem derives an intuitive law of probability known as the law of large numbers from Chebyshev's law. Informally, the law of large numbers says if you repeat an experiment many times, the fraction of the time that an event occurs is very likely to be close to the probability of the event. In particular, we shall prove that for any positive number s, no matter how small, by making the number n independent trials in a sequence of independent trials large enough, we can make the probability that the number X of successes is between np - ns and np + ns as close to 1 as we choose. For example, we can make the probability that the number of successes is within 1% (or 0.1 per cent) of the expected number as close to 1 as we wish.
(a) Show that the probability that |X(x) - np| ≥ sn is no more than p(1 - p)/s2n.
(b) Explain why this means that we can make the probability that X(x) is between np-sn and np + sn as close to 1 as we want by making n large.
What is the probability of there being 2 aces in a single column at the start of a Free Cell game? What about 3 and 4 aces in a single column? Conduct an experiment that validates your findings.
The Sun is shining directly down a missile silo 1000 miles to your south, while at your location, the Sun is 36° from straight overhead. What is the circumference of Myrmidon? What is its radius?
Assignment 6. Without attempting to evaluate the following indefinite integrals, find f'(x) in each case if f(x) is equal to 0∫x(1 + t2)-3 dt And 0∫x^2(1 + t2)-3 dt
Suppose you randomly draw two marbles, without replacement from a bag, containing six green, four red and three black marbles. a) Determine the probability that both marbles are red.
a boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that
Find the curve that passes through the points (3, 2) and has the property that if the tangent line is drawn at any point P on the curve, then the part of the tangent line that lies in the first quadrant is bisected at P.
It currently has a 50 m radius, and the radius is increasing at 0.05 m per second. Your job, as the tanker's captain, is to determine how fast the oil is leaking from the tanker.
What amount today is economically equivalent to $8,900 paid 15 months from now if money is worth 6.6% compounded monthly?
Find the approximate probability that, in the next 50 tosses of a fair coin, the proportion of heads observed will be less than or equal to 0.46. In a large city. 12% of the workforces are unemployed. If 300 people from the workforce are selected at ..
A local hamburger shop sold a combined total of hamburgers and cheeseburgers on Wednesday. There were fewer cheeseburgers sold than hamburgers. How many hamburgers were sold on Wednesday?
a) Give the exact x value of a non-stationary inflection point? b) Give an x value where the function is increasing, but concave down? c) Give an xvalue where the function is decreasing, but concave down?
Calculation involving a change in blood pH. Although the pH of human blood is normally between pH 7.3-7.5, certain situations can take place which bring about "acidosis." Acidosis is defined as an abnormal lowering in the blood pH
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