answer:The time it takes a rat to learn its way through a maze is given as an exponentially distributed random variable with a probability density function (pdf) of ( f(t) = 0.03e^{-0.03t} ) for ( t geq 0 ). To find the probability that a rat will learn its way through a maze in 190 seconds or less, we need to calculate the cumulative distribution function (CDF) for ( t leq 190 ). The CDF for an exponential distribution with rate parameter ( lambda ) is given by: [ F(t) = 1 - e^{-lambda t} ] Here, our rate parameter ( lambda ) is 0.03. Plugging that and ( t = 190 ) into the CDF formula gives us: [ F(190) = 1 - e^{-0.03 times 190} ] Now we can calculate this probability: [ F(190) = 1 - e^{-5.7} ] Using a calculator or software to evaluate the exponential term, we get: [ F(190) = 1 - e^{-5.7} approx 1 - 0.003344 ] Rounded to six decimal places: [ F(190) approx 1 - 0.003344 ] [ F(190) approx 0.996656 ] So, the probability that a rat will learn its way through a maze in 190 seconds or less is approximately 0.996656, or 99.6656%, when rounded to six decimal places.

answer:To find the probability ( P(1 leq x leq 10) ) that the dart hits a number in the interval [1, 10] with the given probability density function (pdf) ( f(x) = frac{x}{50} ), we will integrate the pdf from x = 1 to x = 10. First, let's recall what the pdf tells us: the probability that a dart lands at any specific point is actually zero because we're dealing with a continuous distribution. Instead, the pdf gives us a way to calculate the probability over a range of values. To find the probability that the dart lands within a certain interval, we integrate the pdf over that interval. Let's calculate ( P(1 leq x leq 10) ): [ P(1 leq x leq 10) = int_{1}^{10} frac{x}{50} dx ] Evaluating the integral, we get: [ P(1 leq x leq 10) = left[ frac{x^2}{100} right]_{1}^{10} ] Now, compute the definite integral: [ P(1 leq x leq 10) = frac{10^2}{100} - frac{1^2}{100} ] [ P(1 leq x leq 10) = frac{100}{100} - frac{1}{100} ] [ P(1 leq x leq 10) = 1 - 0.01 ] [ P(1 leq x leq 10) = 0.99 ] Thus, the probability ( P(1 leq x leq 10) ) that the dart lands in the interval [1, 10] is 0.99 or 99%.

answer:For an exponentially distributed random variable with a probability density function ( f(t) = 0.04e^{-0.04t} ), the cumulative distribution function (CDF) representing the probability that the random variable T is less than or equal to a value t is: [ F(t) = 1 - e^{-0.04t} ] To find the probability that a rat requires more than 180 seconds to learn its way through the maze, we are looking for the complement of the probability that the rat finishes in 180 seconds or less, or ( P(T > 180) ). We calculate this by: [ P(T > 180) = 1 - P(T leq 180) ] [ P(T > 180) = 1 - F(180) ] [ P(T > 180) = 1 - (1 - e^{-0.04 times 180}) ] Plugging in the values: [ P(T > 180) = 1 - (1 - e^{-7.2}) ] Now, calculate the exponential term: [ P(T > 180) = 1 - (1 - e^{-7.2}) ] [ P(T > 180) = e^{-7.2} ] Using a calculator or a computer to evaluate ( e^{-7.2} ), we get: [ P(T > 180) approx e^{-7.2} approx 0.000748 ] Rounded to six decimal places, the probability that a rat requires more than 180 seconds to learn its way through the maze is approximately 0.000748 or 0.0748%.

answer:Given that ( x ) is selected from the interval [4, 20] with a uniform distribution, and the probability density function ( f(x) = frac{1}{16} ), we can find the probability that a number selected is in the subinterval [8, 14] by integrating the probability density function over this range. Since ( f(x) ) is constant, this integration is straightforward. The uniform distribution over the interval [4, 20] indicates that the probability is evenly distributed across the interval. With a probability density function ( f(x) = frac{1}{16} ), the total length of the interval is ( 20 - 4 = 16 ), and the area under the curve of the probability density function over the entire interval [4, 20] is 1, which represents a certain event. This confirms that ( f(x) ) is correctly normalized. To find the probability that ( x ) is between 8 and 14, we can simply calculate the length of this subinterval and multiply it by the probability density: [ P(8 leq x leq 14) = int_{8}^{14} f(x)dx ] [ P(8 leq x leq 14) = int_{8}^{14} frac{1}{16}dx ] [ P(8 leq x leq 14) = frac{1}{16} cdot (14 - 8) ] [ P(8 leq x leq 14) = frac{1}{16} cdot 6 ] [ P(8 leq x leq 14) = frac{6}{16} ] [ P(8 leq x leq 14) = frac{3}{8} ] Hence, the probability that a number selected is in the subinterval [8, 14] is ( frac{3}{8} ), or 0.375 when expressed as a decimal.