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Update lab-intro-probability.ipynb #474

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220 changes: 195 additions & 25 deletions lab-intro-probability.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -38,11 +38,39 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 2,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Estimated probability: 0.8835\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"\n",
"import numpy as np\n",
"\n",
"# Parameters\n",
"n_simulations = 100000\n",
"tickets_sold = 460\n",
"show_prob = 0.97\n",
"seats = 450\n",
"\n",
"# Simulate number of passengers who show up\n",
"# Binomial: number of successes (people who show up)\n",
"simulated_showups = np.random.binomial(tickets_sold, show_prob, n_simulations)\n",
"\n",
"# Count how many simulations have <= 450 passengers\n",
"successful_cases = np.sum(simulated_showups <= seats)\n",
"\n",
"# Estimate probability\n",
"probability = successful_cases / n_simulations\n",
"\n",
"print(f\"Estimated probability: {probability:.4f}\")"
]
},
{
Expand Down Expand Up @@ -72,11 +100,36 @@
},
{
"cell_type": "code",
"execution_count": 5,
"execution_count": 3,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Estimated probability: 0.4904\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"\n",
"import numpy as np\n",
"\n",
"# Parameters\n",
"p = 0.3\n",
"n_simulations = 100000\n",
"\n",
"# Simulate number of attempts until first success\n",
"attempts = np.random.geometric(p, n_simulations)\n",
"\n",
"# Count cases where attempts >= 3\n",
"cases = np.sum(attempts >= 3)\n",
"\n",
"# Estimate probability\n",
"probability = cases / n_simulations\n",
"\n",
"print(f\"Estimated probability: {probability:.4f}\")"
]
},
{
Expand Down Expand Up @@ -107,11 +160,29 @@
},
{
"cell_type": "code",
"execution_count": 6,
"execution_count": 4,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Probability of being overwhelmed in one hour: 0.012898\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"\n",
"from scipy.stats import poisson\n",
"\n",
"# Average visits per hour\n",
"lam = 500\n",
"\n",
"# Server overwhelmed if visits > 550\n",
"prob_overwhelmed_hour = 1 - poisson.cdf(550, lam)\n",
"\n",
"print(f\"Probability of being overwhelmed in one hour: {prob_overwhelmed_hour:.6f}\")"
]
},
{
Expand All @@ -123,11 +194,34 @@
},
{
"cell_type": "code",
"execution_count": 7,
"execution_count": 5,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Probability of being overwhelmed in one hour: 0.012898\n",
"Probability of being overwhelmed at least once in 24 hours: 0.267704\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"\n",
"from scipy.stats import poisson\n",
"\n",
"# Average visits per hour\n",
"lam = 500\n",
"\n",
"# Probability of overload in one hour\n",
"p_hour = 1 - poisson.cdf(550, lam)\n",
"\n",
"# Probability of overload at least once in 24 hours\n",
"p_day = 1 - (1 - p_hour) ** 24\n",
"\n",
"print(f\"Probability of being overwhelmed in one hour: {p_hour:.6f}\")\n",
"print(f\"Probability of being overwhelmed at least once in 24 hours: {p_day:.6f}\")"
]
},
{
Expand Down Expand Up @@ -157,10 +251,29 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": []
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Probability customer arrives within 5 minutes: 0.3935\n"
]
}
],
"source": [
"#code here \n",
"\n",
"import numpy as np\n",
"\n",
"lam = 0.1 # rate per minute\n",
"t = 5\n",
"\n",
"prob = 1 - np.exp(-lam * t)\n",
"\n",
"print(f\"Probability customer arrives within 5 minutes: {prob:.4f}\")"
]
},
{
"cell_type": "markdown",
Expand All @@ -173,10 +286,29 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": []
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Probability of no customers for 15 minutes: 0.2231\n"
]
}
],
"source": [
"#code here \n",
"\n",
"import numpy as np\n",
"\n",
"lam = 0.1 # rate per minute\n",
"t = 15\n",
"\n",
"prob = np.exp(-lam * t)\n",
"\n",
"print(f\"Probability of no customers for 15 minutes: {prob:.4f}\")"
]
},
{
"cell_type": "markdown",
Expand All @@ -196,11 +328,28 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 8,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Probability: 0.6827\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"\n",
"from scipy.stats import norm\n",
"\n",
"mu = 150\n",
"sigma = 10\n",
"\n",
"prob = norm.cdf(160, mu, sigma) - norm.cdf(140, mu, sigma)\n",
"\n",
"print(f\"Probability: {prob:.4f}\")"
]
},
{
Expand All @@ -219,11 +368,32 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 9,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Probability of failure within 30 hours: 0.4512\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"\n",
"import numpy as np\n",
"\n",
"# rate\n",
"lam = 1 / 50\n",
"\n",
"# time\n",
"t = 30\n",
"\n",
"# probability\n",
"prob = 1 - np.exp(-lam * t)\n",
"\n",
"print(f\"Probability of failure within 30 hours: {prob:.4f}\")"
]
}
],
Expand All @@ -243,7 +413,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.9"
"version": "3.9.6"
}
},
"nbformat": 4,
Expand Down