ds-probability

Question 1

What is an experiment in probability?

Answer 1

A repeatable procedure with well-defined possible outcomes.

Question 2

What is the sample space in probability?

How do we denote sample sample?

Answer 2

The set (unique values) of all possible outcomes.

We usually denote the sample space by
Ω, sometimes by S.

Question 3

What is an event in probability?

Answer 3

A subset of the sample space.

Question 4

What is a probability function?

Answer 4

a function giving the probability for each outcome

Question 5

What is the probability density function?

Answer 5

A continuous distribution of probabilities.

Question 6

What is a random variable?

Answer 6

A random numerical outcome.

Question 7

What are the 2 conditions that a Probability must satisfy?

Answer 7

The P is a number between 0 and 1.

The sum of all individual probabilities is equal to 1.

Question 8

In probability, what is a stop rule? What is the stop rule in the example below?

Experiment: Toss the coin until the first heads. Report the number of tosses.
Sample space: Ω = {1, 2, 3, . . . }.
Probability function: P (n) = (1 − p)n−1p.

Answer 8

Stopping rule is a rule that tells you when to end a certain process.

In the toy example above the process was flipping a coin
and we stopped after the first heads.

A more practical example is a rule for ending a series
of medical treatments. Such a rule could depend on how well the treatments are working,
how the patient is tolerating them and the probability that the treatments would continue
to be effective. One could ask about the probability of stopping within a certain number of
treatments or the average number of treatments you should expect before stopping.

Question 9

What is the probability of f L and R if they are disjoint?

Answer 9

P (L ∪ R) = P (L) + P (R)

Question 10

If L and R are NOT disjoint, what is the probability of P (L U R)? What is the name of this principle?

Answer 10

P (L ∪ R) = P (L) + P (R) − P (L ∩ R)

It’s called the inclusion-exclusion principle.

Question 11

Please illustrate the following examples using a Venn Diagram:

P (Ac) = 1 − P (A)

P (L ∪ R) = P (L) + P (R)

P (L ∪ R) = P (L) + P (R) − P (L ∩ R)

Answer 11

Question 12

Let A, B and C be the events X is a multiple of 2, 3 and 6 respectively. If P (A) = .6, P (B) = .3 and P (C) = .2 what is P (A or B)?

Answer 12

P (A ∪ B) = P (A) + P (B) − P (A ∩ B) = .6 + .3 − .2 = .7

Question 13

solve the these probabilities.

Question 14

What is the probability of the example attached?

Answer 14

2/3

Question 15

Answer 15

3/4

Question 16

What is conditional probability (P(B∣A))?

Answer 16

The probability that an event will take place given the restrictive assumption that another event has taken place, or that a combination of other events has taken place

P(B∣A)=P(A∩B)/P(A)

Question 17

Enuntiate the Bayes Theorem

Answer 17

Bayes’ theorem P(A∣B)/P(B∣A)=P(A)/P(B)

Question 18

What is P(A∩B) for independent events?

Answer 18

When events are independent, we can use the multiplication rule for independent events, which states that P(A∩B)=P(A)P(B).

Question 19

What is an independent event?

Answer 19

Not contingent or dependent on something else.

Question 20

what is a disjoint event?

Answer 20

Having no members in common; having an intersection equal to the empty set.

Question 21

Assume that P(A) = 0.3, P(B/A) = 0.75, and P(C/A∩B) = 0.20. What is P(A∩B∩C)?

Answer 21

9/200