You're a data scientist analyzing website traffic. You have data on the average time users spend on a page, which is 2 minutes with a standard deviation of 30 seconds. You want to estimate the probability that the average time spent on the page by a random sample of 100 users is between 1 minute 55 seconds and 2 minutes 5 seconds. What concept would be most suitable for this calculation?
Conditional Probability
Central Limit Theorem
Bayes' Theorem
Law of Large Numbers
A fair coin is tossed three times. What is the probability of getting at least two heads?
1/8
1/2
3/8
7/8
In a large dataset of customer purchase amounts, the average purchase is $50 with a standard deviation of $20. If we take 100 random samples of size 25 from this dataset, what will be the standard deviation of the distribution of these sample means?
$2
$4
$20
$10
Suppose a diagnostic test for a certain disease has a 95% sensitivity and 90% specificity. It means that the test correctly identifies 95% of people with the disease and correctly identifies 90% of people without the disease. If the prevalence of the disease in a population is 1%, what is the probability that a person who tests positive actually has the disease?
0.095
0.090
0.086
0.500
In a normal distribution, what percentage of data falls within three standard deviations of the mean?
100%
95.45%
99.73%
68.27%
In the context of machine learning, how does the Law of Large Numbers relate to training data?
It proves that a complex model will always outperform a simpler model given enough data.
The Law of Large Numbers is irrelevant to machine learning; it's a purely statistical concept.
A model trained on a larger, more representative dataset is likely to generalize better to unseen data.
The Law of Large Numbers dictates the optimal learning rate for a machine learning model.
If the conditional probability P(Y=y | X=x) is equal to the marginal probability P(Y=y) for all values of x and y, what does this imply about the relationship between X and Y?
X and Y are dependent.
X and Y are independent.
X and Y are uniformly distributed.
X and Y are mutually exclusive.
In a certain city, 70% of the population owns a car and 40% owns a bike. Among the car owners, 30% also own a bike. If a person is randomly selected from the city, what is the probability that they own both a car and a bike?
0.58
0.21
0.28
0.12
How is the marginal probability distribution of X obtained from the joint probability distribution of X and Y?
By multiplying the joint probabilities by the marginal probabilities of Y
By dividing the joint probabilities by the marginal probabilities of Y
By subtracting the joint probabilities from 1
By summing the joint probabilities over all possible values of Y
Events A and B are independent. The probability of event A is 0.3. The probability of event B is 0.6. What is the probability of both A and B occurring?
Cannot be determined from the given information.
0.50
0.18
0.90