document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166169>Question 1166169</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The number of members arriving at a gym follows a Poisson distribution with a mean of 2 members per minute.\r<BR>\n" );
document.write( "\n" );
document.write( "(a) What is the probability that 3 members arrive in 30 seconds?<BR>\n" );
document.write( "(b) Determine the probability (in percentage) that the lady at the front entrance can have a 2 minute uninterrupted coffee break?\r<BR>\n" );
document.write( "\n" );
document.write( "The average time a member trains at the gym is 80 minutes with a standard deviation of 10 minutes. Assuming a normal distribution, determine the probability that a randomly selected member trains:\r<BR>\n" );
document.write( "\n" );
document.write( "(c) for longer than an hour.<BR>\n" );
document.write( "(d) between 90 and 105 minutes. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852868 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852868\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "from scipy.stats import poisson, norm\r<BR>\n" );
document.write( "\n" );
document.write( "# (a) Probability that 3 members arrive in 30 seconds (0.5 min)<BR>\n" );
document.write( "# Mean mu_a = 2 members/min * 0.5 min = 1<BR>\n" );
document.write( "mu_a = 1<BR>\n" );
document.write( "P_a = poisson.pmf(3, mu_a)\r<BR>\n" );
document.write( "\n" );
document.write( "# (b) Probability that 0 members arrive in 2 minutes (uninterrupted break)<BR>\n" );
document.write( "# Mean mu_b = 2 members/min * 2 min = 4<BR>\n" );
document.write( "mu_b = 4<BR>\n" );
document.write( "P_b_decimal = poisson.pmf(0, mu_b)<BR>\n" );
document.write( "P_b_percentage = P_b_decimal * 100\r<BR>\n" );
document.write( "\n" );
document.write( "# (c) Probability that a member trains for longer than an hour (60 minutes)<BR>\n" );
document.write( "# T > 60. Mu = 80, Sigma = 10. Z = (60 - 80) / 10 = -2.0<BR>\n" );
document.write( "Z_c = -2.0<BR>\n" );
document.write( "# P(Z > -2.0) = 1 - P(Z <= -2.0)<BR>\n" );
document.write( "P_c = 1 - norm.cdf(Z_c)\r<BR>\n" );
document.write( "\n" );
document.write( "# (d) Probability that a member trains between 90 and 105 minutes<BR>\n" );
document.write( "# Z_90 = (90 - 80) / 10 = 1.0<BR>\n" );
document.write( "Z_90 = 1.0<BR>\n" );
document.write( "# Z_105 = (105 - 80) / 10 = 2.5<BR>\n" );
document.write( "Z_105 = 2.5<BR>\n" );
document.write( "# P(1.0 < Z < 2.5) = P(Z < 2.5) - P(Z < 1.0)<BR>\n" );
document.write( "P_d = norm.cdf(Z_105) - norm.cdf(Z_90)\r<BR>\n" );
document.write( "\n" );
document.write( "print(f\"P(a): {P_a}\")<BR>\n" );
document.write( "print(f\"P(b) decimal: {P_b_decimal}\")<BR>\n" );
document.write( "print(f\"P(b) percentage: {P_b_percentage}\")<BR>\n" );
document.write( "print(f\"P(c): {P_c}\")<BR>\n" );
document.write( "print(f\"P(d): {P_d}\")\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "P(a): 0.06131324019524039<BR>\n" );
document.write( "P(b) decimal: 0.01831563888873418<BR>\n" );
document.write( "P(b) percentage: 1.8315638888734178<BR>\n" );
document.write( "P(c): 0.9772498680518208<BR>\n" );
document.write( "P(d): 0.15244558860568092\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "This is a comprehensive probability question that tests your understanding of both **Poisson** and **Normal** distributions. Here are the step-by-step solutions for each part.\r<BR>\n" );
document.write( "\n" );
document.write( "## Member Arrivals (Poisson Distribution)\r<BR>\n" );
document.write( "\n" );
document.write( "The number of members arriving follows a Poisson distribution with a mean rate $\lambda = 2$ members per minute. The formula for the Poisson probability mass function is:<BR>\n" );
document.write( "$$P(X=k) = \frac{e^{-\lambda t} (\lambda t)^k}{k!}$$<BR>\n" );
document.write( "where $\lambda t$ is the expected number of arrivals in the time interval $t$.\r<BR>\n" );
document.write( "\n" );
document.write( "### (a) What is the probability that 3 members arrive in 30 seconds?\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Calculate the mean ($\lambda t$) for the given interval:**<BR>\n" );
document.write( "    The time interval $t$ is $30$ seconds, which is $0.5$ minutes.<BR>\n" );
document.write( "    $$\lambda t = (2 \text{ members/min}) \times (0.5 \text{ min}) = 1$$<BR>\n" );
document.write( "    The mean number of members arriving in $30$ seconds is $\mu = 1$.\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Calculate $P(X=3)$:**<BR>\n" );
document.write( "    Using the Poisson formula with $k=3$ and $\lambda t=1$:<BR>\n" );
document.write( "    $$P(X=3) = \frac{e^{-1} (1)^3}{3!} = \frac{e^{-1}}{6}$$<BR>\n" );
document.write( "    $$P(X=3) \approx \mathbf{0.0613}$$\r<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( "### (b) Determine the probability (in percentage) that the lady at the front entrance can have a 2 minute uninterrupted coffee break?\r<BR>\n" );
document.write( "\n" );
document.write( "An uninterrupted coffee break means **zero** members arrive during the $2$ minutes.\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Calculate the mean ($\lambda t$) for the given interval:**<BR>\n" );
document.write( "    The time interval $t$ is $2$ minutes.<BR>\n" );
document.write( "    $$\lambda t = (2 \text{ members/min}) \times (2 \text{ min}) = 4$$<BR>\n" );
document.write( "    The mean number of members arriving in $2$ minutes is $\mu = 4$.\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Calculate $P(X=0)$:**<BR>\n" );
document.write( "    Using the Poisson formula with $k=0$ and $\lambda t=4$:<BR>\n" );
document.write( "    $$P(X=0) = \frac{e^{-4} (4)^0}{0!} = e^{-4}$$<BR>\n" );
document.write( "    $$P(X=0) \approx 0.018315$$\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Convert to percentage:**<BR>\n" );
document.write( "    $0.018315 \times 100\% \approx \mathbf{1.83\%}$\r<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( "## Member Training Time (Normal Distribution)\r<BR>\n" );
document.write( "\n" );
document.write( "The training time $T$ follows a Normal distribution with mean $\mu = 80$ minutes and standard deviation $\sigma = 10$ minutes.\r<BR>\n" );
document.write( "\n" );
document.write( "We use the **Z-score** formula $Z = \frac{T - \mu}{\sigma}$ to standardize the values.\r<BR>\n" );
document.write( "\n" );
document.write( "### (c) for longer than an hour.\r<BR>\n" );
document.write( "\n" );
document.write( "We want to find $P(T > 1 \text{ hour})$. Since $1 \text{ hour} = 60 \text{ minutes}$: $P(T > 60)$.\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Calculate the Z-score for $T=60$:**<BR>\n" );
document.write( "    $$Z_{60} = \frac{60 - 80}{10} = \frac{-20}{10} = -2.0$$\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    We look for $P(Z > -2.0)$. Since the Normal distribution is symmetric, $P(Z > -z) = P(Z < z)$.<BR>\n" );
document.write( "    $$P(Z > -2.0) = 1 - P(Z \le -2.0)$$<BR>\n" );
document.write( "    Using the standard normal table or calculator for $Z=-2.0$:<BR>\n" );
document.write( "    $$P(Z > -2.0) \approx \mathbf{0.9772}$$\r<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( "### (d) between 90 and 105 minutes.\r<BR>\n" );
document.write( "\n" );
document.write( "We want to find $P(90 < T < 105)$.\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Calculate the Z-scores for $T=90$ and $T=105$:**<BR>\n" );
document.write( "    $$Z_{90} = \frac{90 - 80}{10} = \frac{10}{10} = \mathbf{1.0}$$<BR>\n" );
document.write( "    $$Z_{105} = \frac{105 - 80}{10} = \frac{25}{10} = \mathbf{2.5}$$\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Find the probability:**<BR>\n" );
document.write( "    We look for $P(1.0 < Z < 2.5)$.<BR>\n" );
document.write( "    $$P(1.0 < Z < 2.5) = P(Z < 2.5) - P(Z < 1.0)$$<BR>\n" );
document.write( "    Using the standard normal table or calculator:<BR>\n" );
document.write( "    $$\begin{aligned} P(Z < 2.5) &\approx 0.99379 \\ P(Z < 1.0) &\approx 0.84134 \end{aligned}$$<BR>\n" );
document.write( "    $$P(1.0 < Z < 2.5) = 0.99379 - 0.84134 = \mathbf{0.15245}$$ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );