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218 changes: 192 additions & 26 deletions lab-intro-probability.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -38,11 +38,31 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 2,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that at least 10 passengers don't show up is: 0.8845\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"\n",
"from scipy.stats import binom\n",
"\n",
"# n: total number of tickets sold\n",
"n = 460\n",
"# p: probability of a no-show for each passenger\n",
"p = 0.03\n",
"\n",
"# Compute the probability of at least 10 passengers not showing up\n",
"probability_at_least_10_no_shows = binom.sf(9, n, p) # sf is the survival function, gives P(X > 9)\n",
"\n",
"print(f\"The probability that at least 10 passengers don't show up is: {probability_at_least_10_no_shows:.4f}\")\n"
]
},
{
Expand Down Expand Up @@ -72,11 +92,29 @@
},
{
"cell_type": "code",
"execution_count": 5,
"execution_count": 4,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability of needing at least three attempts is: 0.4900\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"from scipy.stats import geom\n",
"\n",
"# Probability of success on a single attempt\n",
"p = 0.3\n",
"\n",
"# Calculate the probability of needing at least three attempts\n",
"# This is equivalent to 1 - P(X <= 2), where X is the attempt on which the first success occurs\n",
"probability_at_least_three_attempts = 1 - geom.cdf(2, p)\n",
"\n",
"print(f\"The probability of needing at least three attempts is: {probability_at_least_three_attempts:.4f}\")"
]
},
{
Expand Down Expand Up @@ -109,9 +147,26 @@
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability of the website server being overwhelmed is: 0.0129\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"from scipy.stats import poisson\n",
"\n",
"# Average rate (λ) of visits per hour\n",
"lambda_ = 500\n",
"\n",
"# Calculate the probability of more than 550 visits\n",
"probability_overwhelmed = 1 - poisson.cdf(550, lambda_)\n",
"\n",
"print(f\"The probability of the website server being overwhelmed is: {probability_overwhelmed:.4f}\")"
]
},
{
Expand All @@ -123,11 +178,32 @@
},
{
"cell_type": "code",
"execution_count": 7,
"execution_count": 16,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability of the website server being overwhelmed at least once in 24 hours is: 0.2677\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"\n",
"from scipy.stats import poisson\n",
"\n",
"# Average rate (λ) of visits per hour\n",
"lambda_ = 500\n",
"\n",
"# Probability of being overwhelmed in one hour\n",
"probability_overwhelmed_hourly = 1 - poisson.cdf(550, lambda_)\n",
"\n",
"# Probability of being overwhelmed at least once in 24 hours\n",
"probability_overwhelmed_daily = 1 - (1 - probability_overwhelmed_hourly) ** 24\n",
"\n",
"print(f\"The probability of the website server being overwhelmed at least once in 24 hours is: {probability_overwhelmed_daily:.4f}\")"
]
},
{
Expand Down Expand Up @@ -157,10 +233,31 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": []
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that the next customer will arrive within the next 5 minutes is: 0.3935\n"
]
}
],
"source": [
"import math\n",
"\n",
"# lambda: rate of arrivals per minute\n",
"lambda_rate = 1 / 10\n",
"\n",
"# t: time in minutes (within which we're calculating the probability of arrival)\n",
"t = 5\n",
"\n",
"# Calculate the probability using the exponential distribution CDF formula\n",
"probability_within_5_minutes = 1 - math.exp(-lambda_rate * t)\n",
"\n",
"print(f\"The probability that the next customer will arrive within the next 5 minutes is: {probability_within_5_minutes:.4f}\")"
]
},
{
"cell_type": "markdown",
Expand All @@ -173,10 +270,31 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 20,
"metadata": {},
"outputs": [],
"source": []
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability an employee can take a break, with no customers arriving for 15 minutes, is: 0.2231\n"
]
}
],
"source": [
"import math\n",
"\n",
"# lambda: rate of arrivals per minute\n",
"lambda_rate = 1 / 10\n",
"\n",
"# t: time in minutes (no customer arrival period)\n",
"t = 15\n",
"\n",
"# Calculate the probability using the formula for no arrivals\n",
"probability_no_arrival_15_minutes = math.exp(-lambda_rate * t)\n",
"\n",
"print(f\"The probability an employee can take a break, with no customers arriving for 15 minutes, is: {probability_no_arrival_15_minutes:.4f}\")"
]
},
{
"cell_type": "markdown",
Expand All @@ -196,11 +314,29 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 22,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that a randomly selected bird weighs between 140 and 160 grams is: 0.6827\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"from scipy.stats import norm\n",
"\n",
"# Mean and standard deviation\n",
"mu = 150\n",
"sigma = 10\n",
"\n",
"# Calculate the probability\n",
"probability = norm.cdf(160, mu, sigma) - norm.cdf(140, mu, sigma)\n",
"\n",
"print(f\"The probability that a randomly selected bird weighs between 140 and 160 grams is: {probability:.4f}\")"
]
},
{
Expand All @@ -219,17 +355,47 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 24,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that the component fails within the first 30 hours is: 0.4512\n"
]
}
],
"source": [
"#code here"
"#code here\n",
"import math\n",
"\n",
"# Mean lifetime\n",
"mean_lifetime = 50\n",
"\n",
"# Rate (lambda)\n",
"lambda_rate = 1 / mean_lifetime\n",
"\n",
"# Desired time period\n",
"t = 30\n",
"\n",
"# Probability calculation for P(T <= 30)\n",
"probability_failure_within_30_hours = 1 - math.exp(-lambda_rate * t)\n",
"\n",
"print(f\"The probability that the component fails within the first 30 hours is: {probability_failure_within_30_hours:.4f}\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
Expand All @@ -243,9 +409,9 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.9"
"version": "3.12.4"
}
},
"nbformat": 4,
"nbformat_minor": 2
"nbformat_minor": 4
}