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Lab 21 Ben #170

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208 changes: 182 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 +38,38 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 1,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that there are enough seats for all passengers is approximately 0.8505 or 85.05%.\n"
]
}
],
"source": [
"#code here"
"import scipy.stats as stats\n",
"\n",
"seats = 450\n",
"tickets_sold = 460\n",
"p_missing = 0.03\n",
"p_showing = 1 - p_missing\n",
"\n",
"# Calculate mean and standard deviation for the normal approximation\n",
"mean = tickets_sold * p_showing\n",
"variance = tickets_sold * p_showing * p_missing\n",
"std_dev = variance ** 0.5\n",
"\n",
"# Calculate the Z-score for 450 passengers showing up\n",
"z = (seats - mean) / std_dev\n",
"\n",
"# Calculate the cumulative probability\n",
"probability = stats.norm.cdf(z)\n",
"\n",
"# Output the result\n",
"print(f\"The probability that there are enough seats for all passengers is approximately {probability:.4f} or {probability * 100:.2f}%.\")\n"
]
},
{
Expand Down Expand Up @@ -72,11 +99,26 @@
},
{
"cell_type": "code",
"execution_count": 5,
"execution_count": 3,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that the representative needs to make at least three attempts is approximately 0.4900 or 49.00%.\n"
]
}
],
"source": [
"#code here"
"p_success = 0.3\n",
"p_failure = 1 - p_success\n",
"\n",
"# Probability of needing at least three attempts\n",
"probability_at_least_three_attempts = p_failure ** 2\n",
"\n",
"# Output the result\n",
"print(f\"The probability that the representative needs to make at least three attempts is approximately {probability_at_least_three_attempts:.4f} or {probability_at_least_three_attempts * 100:.2f}%.\")\n"
]
},
{
Expand Down Expand Up @@ -107,11 +149,26 @@
},
{
"cell_type": "code",
"execution_count": 6,
"execution_count": 5,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability of the website server being overwhelmed is approximately 0.0129 or 1.29%.\n"
]
}
],
"source": [
"#code here"
"lambda_visits = 500\n",
"server_capacity = 550\n",
"\n",
"# Calculate the probability of the server being overwhelmed\n",
"probability_overwhelmed = 1 - stats.poisson.cdf(server_capacity, lambda_visits)\n",
"\n",
"# Output the result\n",
"print(f\"The probability of the website server being overwhelmed is approximately {probability_overwhelmed:.4f} or {probability_overwhelmed * 100:.2f}%.\")\n"
]
},
{
Expand All @@ -125,9 +182,26 @@
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability of the website server being overwhelmed at some point during a day is approximately 0.0000 or 0.00%.\n"
]
}
],
"source": [
"#code here"
"lambda_visits_per_hour = 500\n",
"hours_in_a_day = 24\n",
"total_visits = lambda_visits_per_hour * hours_in_a_day # 12000 visits in a day\n",
"server_capacity = 550 * hours_in_a_day # 13200 visits\n",
"\n",
"# Calculate the probability of being overwhelmed at least once during the day\n",
"probability_overwhelmed_daily = 1 - stats.poisson.cdf(server_capacity, total_visits)\n",
"\n",
"# Output the result\n",
"print(f\"The probability of the website server being overwhelmed at some point during a day is approximately {probability_overwhelmed_daily:.4f} or {probability_overwhelmed_daily * 100:.2f}%.\")"
]
},
{
Expand Down Expand Up @@ -157,10 +231,31 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 11,
"metadata": {},
"outputs": [],
"source": []
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that the next customer will arrive within the next 5 minutes is approximately 0.3935 or 39.35%.\n"
]
}
],
"source": [
"import math\n",
"\n",
"# Given data\n",
"lambda_time = 10 # Average time between arrivals in minutes\n",
"mu = 1 / lambda_time # Rate of arrivals in customers per minute\n",
"t = 5 # Time period of interest in minutes\n",
"\n",
"# Calculate the probability that the next customer arrives within the next 5 minutes\n",
"probability_next_customer = 1 - math.exp(-mu * t)\n",
"\n",
"# Output the result\n",
"print(f\"The probability that the next customer will arrive within the next 5 minutes is approximately {probability_next_customer:.4f} or {probability_next_customer * 100:.2f}%.\")\n"
]
},
{
"cell_type": "markdown",
Expand All @@ -173,10 +268,29 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 13,
"metadata": {},
"outputs": [],
"source": []
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that an employee can take a break is approximately 0.6065 or 60.65%.\n"
]
}
],
"source": [
"lambda_time = 10 # Average time between arrivals in minutes\n",
"mu = 1 / lambda_time # Rate of arrivals in customers per minute\n",
"t_break = 5 # Time for a break in minutes\n",
"t_no_customer = 15 # Time without a customer in minutes\n",
"\n",
"# Calculate the probability of no customer arriving in the next 5 minutes\n",
"probability_no_customer_next_5 = math.exp(-mu * t_break)\n",
"\n",
"# Output the result\n",
"print(f\"The probability that an employee can take a break is approximately {probability_no_customer_next_5:.4f} or {probability_no_customer_next_5 * 100:.2f}%.\")\n"
]
},
{
"cell_type": "markdown",
Expand All @@ -196,11 +310,30 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 15,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that a randomly selected bird weighs between 140 and 160 grams is approximately 0.6827 or 68.27%.\n"
]
}
],
"source": [
"#code here"
"mean_weight = 150 # Mean in grams\n",
"std_dev_weight = 10 # Standard deviation in grams\n",
"\n",
"# Calculate probabilities\n",
"probability_140 = stats.norm.cdf(140, loc=mean_weight, scale=std_dev_weight)\n",
"probability_160 = stats.norm.cdf(160, loc=mean_weight, scale=std_dev_weight)\n",
"\n",
"# Probability that weight is between 140 and 160 grams\n",
"probability_between = probability_160 - probability_140\n",
"\n",
"# Output the result\n",
"print(f\"The probability that a randomly selected bird weighs between 140 and 160 grams is approximately {probability_between:.4f} or {probability_between * 100:.2f}%.\")"
]
},
{
Expand All @@ -219,17 +352,40 @@
},
{
"cell_type": "code",
"execution_count": null,
"execution_count": 17,
"metadata": {},
"outputs": [],
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The probability that the component fails within the first 30 hours is approximately 0.4512 or 45.12%.\n"
]
}
],
"source": [
"#code here"
"mean_lifetime = 50 # Mean lifetime in hours\n",
"lambda_rate = 1 / mean_lifetime # Rate parameter\n",
"time_of_interest = 30 # Time in hours\n",
"\n",
"# Calculate the probability that the component fails within the first 30 hours\n",
"probability_failure_within_30 = 1 - math.exp(-lambda_rate * time_of_interest)\n",
"\n",
"# Output the result\n",
"print(f\"The probability that the component fails within the first 30 hours is approximately {probability_failure_within_30:.4f} or {probability_failure_within_30 * 100:.2f}%.\")"
]
},
{
"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 +399,9 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.9"
"version": "3.12.4"
}
},
"nbformat": 4,
"nbformat_minor": 2
"nbformat_minor": 4
}