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Dependent Event
Problem 113: A sample of two balls is drawn from an urn containing two white balls and three red balls. Are the events A = “the sample contains at least one white ball” and B = “the sample contains balls of both colors” independent? Solution: If events and are independent, then and . 1. Calculate…
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Intercepts of Rational Functions
Problem 112: Given the function (1) find (a) the y-intercept, and (b) x- intercept. Solution: First, we need to know the definition of the y-intercept and x-intercept. That is, the y-intercept is given by the point and the x-intercept is given by the point . (a) The y-intercept is then (2) That is,…
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Even and Odd Functions
Problem 111: Determine if is even or odd: (1) Solution: Before we go ahead and solve the problem, let’s remind ourselves that a function is even if . Moreover, a function is odd if . That is, (2) Since , we say that is an odd function.
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Derivative using Logarithmic Differentiation
Problem 110: Find the derivative of using logarithmic differentiation when . Solution: We will take the derivative of using logarithmic properties. That is, (1) Differentiating both side we get (2) That is, (3)
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Derivative with Logarithmic Differentiation
Problem 109: Find the derivative of using logarithmic differentiation when . Solution: We will take the derivative of using logarithmic properties. That is, (1) Differentiating both side we get (2) That is, (3)
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Conditional Probability
Problem 108: Heads and Tails Three ordinary quarters and a fake quarter with two heads are placed in a hat. One quarter is selected at random and tossed twice. If the outcome is “HH”, what is the probability that the fake quarter was selected? Solution: I believe that sometimes is better to start from the…
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Probability with Equally Likely Outcomes
Problem 107: Suppose that a cruise ship returns to the United States from the FarEast. Unknown to anyone, 4 of its 600 passengers have contracted a rare disease. Suppose that the Public Health Service screens 20 passengers, selected at random, to see whether the disease is present aboard ship. What is the probability that the…
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Volume of an Sphere and Cone
Problem 106: Find the volume of a half sphere (radius 5 yds) on the base of an inverted cone with the same radius and height of 7 yds. Solution: We will solve this problem by finding the volume of the shapes separately and then add them. The volume of an sphere is given by ….
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Probability with Equally Likely Outcomes
Problem 105: An urn contains eight white balls and two green balls. A sample of three balls is selected at random. (a) What is the probability that the sample contains only white balls. (b) What is the probability that the sample contains at least one green ball? Solution: The probability of an event, E, is…
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Th Binomial Theorem
Problem 104: Expand . Solution: If we want to expand this, we can either multiply it out 6 times or use the binomial theorem. We will use the latter here. That is, Note that by using the binomial theorem, we just had to calculate the coefficients.
One Problem at a Time
