Question

The length of time for students to complete a 1-hour timed standardized mathematics test has a probability density function f(t) = 1.5t^2 + t, 0 ≤ t ≤ 1, where t is the time in hours. (Round your answers to three decimal places.) (a) Find the probability that the time it takes a randomly selected student to complete the test is more than 51 minutes. (b) Find the probability that the time it takes a randomly selected student to complete the test is more than 15 minutes given that it is less than 51 minutes. 

          The length of time for students to complete a 1-hour timed standardized mathematics test has a probability density function f(t) = 1.5t^2 + t, 0 ≤ t ≤ 1, where t is the time in hours. (Round your answers to three decimal places.)

(a) Find the probability that the time it takes a randomly selected student to complete the test is more than 51 minutes.

(b) Find the probability that the time it takes a randomly selected student to complete the test is more than 15 minutes given that it is less than 51 minutes.
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Added by Marcus R.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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The length of time for students to complete a 1-hour timed standardized mathematics test has a probability density function f(t) = 1.5t^2 + t, 0 ≤ t ≤ 1, where t is the time in hours. (Round your answers to three decimal places.) (a) Find the probability that the time it takes a randomly selected student to complete the test is more than 51 minutes. (b) Find the probability that the time it takes a randomly selected student to complete the test is more than 15 minutes given that it is less than 51 minutes.
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Transcript

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00:01 Okay, so let's get started with a.
00:04 Our probability density function, f -o -t, is 3 -2t2t, with t belonging to the interval 0 -1.
00:16 Now, what is the probability that a randomly selected student is going to complete the test in more than 55 minutes? well this probability p is gonna be equal to okay 51 minutes is 51 over 60 because t is in hours so here we are gonna have an integral between 51 over 60 and 1 of f of t in the t now f of t is this parabola here is this barabula here and we know that an antiderivative is going to be t cubed over two plus t squared over two and we need to evaluate this guy between 51 over 60 and one perfect so what we get well we get the evaluation at 1 is just 1 minus now we are going to have 1 half multiplied by 50 51 cubed over 60 cubed plus 51 squared over 60 squared perfect.
01:40 So as we can see the first part of our exercise was easy.
01:46 Now for b well our randomly selected student is gonna take more than 50 minutes and less than 51 minutes to complete the test.
01:57 So here our probability is going to be what? well easy...
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