Probability - discrete sample space, events, classical, relative frequency and axiomatic approaches, conditional probability, independence, Bayes' theorem - One Line Questions
1.
If P(A) = 0.6, P(A') = 0.4. If P(B|A) = 0.5, what is P(A|B) if P(B) = 0.7? —
(0.6 * 0.5) / 0.7
2.
Consider rolling a fair six-sided die. What is the sample space? —
{1, 2, 3, 4, 5, 6}
3.
Let S = {1, 2, 3, 4, 5, 6} be the sample space for rolling a die. What is the event of rolling an even number? —
{2, 4, 6}
4.
If you flip a fair coin twice, what is the sample space? —
{HH, HT, TH, TT}
5.
According to the axiomatic approach, what is the probability of the sample space S? —
1
6.
If P(A) = 0.8 and P(B) = 0.7, what is the minimum possible value for P(A ∩ B)? —
0.1
7.
If P(A) = 0.3, P(B) = 0.6, and P(A|B) = 0.2, what is P(A ∩ B)? —
0.2 * 0.6
8.
Consider the events A: 'It is raining' and B: 'The ground is wet'. If P(A) = 0.3, P(B) = 0.4, and P(B|A) = 0.8, what is P(A ∩ B)? —
0.3 * 0.8
9.
If P(A) = 0.5 and P(B) = 0.6, and P(A ∩ B) = 0.4, what is P(A|B)? —
0.4 / 0.6
10.
A coin is tossed 100 times, and heads appears 45 times. What is the relative frequency estimate of the probability of getting heads? —
0.45
11.
If P(A) = 0.6 and P(B) = 0.3, and A and B are independent, what is P(A ∩ B)? —
0.18
12.
Suppose a medical test for a disease is 99% accurate (detects the disease if present) and has a 1% false positive rate (indicates disease if not present). If 0.5% of the population has the disease, what is the probability that a person has the disease given a positive test result? —
Approximately 0.33
13.
If P(A) = 0.7 and P(B) = 0.4, and P(A ∩ B) = 0.2, what is P(A U B)? —
0.9
14.
What is the probability of drawing an ace from a standard deck of 52 cards in a single draw, using the classical approach? —
4/52
15.
What is the probability that a randomly selected integer from 1 to 30 is divisible by either 4 or 6? —
9/30
16.
What is the probability of drawing a red card or a face card (King, Queen, Jack) from a standard deck of 52 cards? —
26/52 + 12/52 - 6/52
17.
Consider the experiment of selecting a letter from the word 'STATISTICS'. What is the probability of selecting the letter 'S'? —
4/10
18.
Let S = {1, 2, 3, ..., 10}. Let E be the event that a number chosen from S is divisible by 3. What is P(E)? —
3/10
19.
What is the probability of rolling a sum of 8 with two fair dice? —
5/36
20.
What is the probability of selecting a vowel from the letters of the word 'EQUATION'? —
5/8
21.
Bayes' theorem provides a way to: —
Update the probability of a hypothesis based on new evidence
22.
Which approach to probability is based on a set of axioms or postulates? —
Axiomatic Approach
23.
Which approach to probability is most suitable for situations involving personal beliefs or opinions? —
Subjective Approach (though not explicitly detailed in the topic, it's a contrasting concept)
24.
If two events A and B are independent, then A' and B' are also: —
Independent
25.
What is a set of all possible outcomes of a random experiment called? —
Sample Space
26.
In the axiomatic approach, what is the probability of any event P(E) guaranteed to be? —
Between 0 and 1, inclusive
27.
In the classical approach to probability, what is the probability of an event E assumed to be? —
Number of favorable outcomes / Total number of outcomes
28.
What is the probability of event A occurring given that event B has NOT occurred, P(A|B')? —
P(A ∩ B') / P(B')
29.
The formula for conditional probability P(A|B) is: —
P(A ∩ B) / P(B)
30.
If events A and B are independent, which of the following is true? —
P(A ∩ B) = P(A) * P(B)
31.
If events A and B are mutually exclusive, what is P(A ∩ B)? —
0
32.
For conditional probability P(A|B) to be defined, what must be true? —
P(B) > 0
33.
Bayes' theorem is mathematically expressed as: —
P(A|B) = P(B|A)P(A) / P(B)
34.
In a discrete sample space, what is the probability of an event E that is composed of a finite number of sample points {e1, e2, ..., ek}? —
P(E) = P(e1) + P(e2) + ... + P(ek)
35.
If E1 and E2 are mutually exclusive events, what is P(E1 U E2) according to the axiomatic approach? —
P(E1) + P(E2)
36.
Which axiom of probability states that probabilities are non-negative? —
P(E) >= 0 for any event E
37.
In Bayes' theorem, P(B|A) is often referred to as the: —
Likelihood
38.
In Bayes' theorem, P(A) is referred to as the: —
Prior probability
39.
In Bayes' theorem, P(A|B) is called the: —
Posterior probability
40.
What is conditional probability denoted as P(A|B)? —
Probability of A occurring given that B has occurred
41.
Which of the following is NOT a property of probability? —
Probability of an impossible event is 1.
42.
Which of the following represents a subset of the sample space? —
Event
43.
What does the relative frequency approach to probability estimate? —
The probability of an event based on past occurrences
44.
The relative frequency approach to probability is often used when: —
It's impossible to determine equally likely outcomes
45.
What does it mean for two events A and B to be independent? —
The occurrence of A does not affect the probability of B
46.
If P(A|B) = P(A), what can be concluded about events A and B? —
They are independent
47.
When is the classical approach to probability applicable? —
When all outcomes are equally likely and finite
48.
Let A be the event of getting a sum of 7 when rolling two dice, and B be the event of getting a 4 on the first die. Are A and B independent? —
No, because P(A|B) != P(A)
49.
If P(A) = 0.4, P(B) = 0.5, and P(A U B) = 0.7, are events A and B independent? —
Yes, because P(A ∩ B) = 0.2, and P(A) * P(B) = 0.2