A spam filter correctly identifies 95% of spam emails. However, it also flags 2% of legitimate emails as spam. If 1% of all emails are actually spam, what is the probability that an email flagged as spam is actually spam?
95%
50%
32%
2%
A machine produces bolts with diameters normally distributed, a mean of 10mm, and a standard deviation of 0.2mm. What percentage of bolts will have a diameter between 9.7mm and 10.3mm?
68.27%
95.45%
99.73%
86.64%
You're analyzing the average height of trees in a forest. You take multiple samples of 50 trees each. According to the Central Limit Theorem, what can you infer about the distribution of the sample means of these tree heights?
The distribution of sample means will be identical to the distribution of individual tree heights.
The distribution of sample means will be skewed right.
The Central Limit Theorem cannot be applied to this situation.
The distribution of sample means will be approximately normal.
You roll two six-sided dice. Are the events "getting a sum of 7" and "getting doubles" independent events?
Yes
No
The time until a radioactive particle decays is modeled by an exponential distribution. If the average decay time is 10 seconds, what is the median decay time?
10 seconds
6.93 seconds
5 seconds
14.43 seconds
A fair coin is tossed three times. What is the probability of getting at least two heads?
7/8
1/2
3/8
1/8
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?
Law of Large Numbers
Conditional Probability
Bayes' Theorem
Central Limit Theorem
The time between arrivals of customers at a store is exponentially distributed with a mean of 5 minutes. What is the probability that the next customer will arrive in less than 2 minutes?
0.3297
0.4721
0.5279
0.6703
If two random variables X and Y are independent, what can be said about their joint probability distribution?
It is always a uniform distribution.
It is equal to the product of their marginal distributions.
It is equal to the sum of their marginal distributions.
It cannot be determined from their marginal distributions.
The lifetime of a certain component follows a gamma distribution with a shape parameter of 3 and a rate parameter of 0.2. What is the variance of the component's lifetime?
15
75
45
7.5