TASK 1

Question 1

How would you define a random phenomenon?

Answer 1

We call phenomenon random if individual outcomes are uncertain but there in nonetheless a regular distribution of outcomes in a large number of repetitions. “Random” in statistics is not a synonym for “haphazard” but a description of a kind of order that emerges only in the long run.

Question 2

What is a probabilistic experiment? Offer examples.

Answer 2

A probabilistic experiment is an occurrence such as the tossing of a coin, rolling of a die, etc. in which the complexity of the underlying system leads to an outcome that cannot be known ahead of time.

Question 3

How would you define the sample space in a probabilistic experiment? Offer examples.

Answer 3

_sum of all possible outcomes in a probability experiment (S) _ Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly 1 _EXAMPLE: Experiment Rolling a die once: Sample space S = {1, 2, 3, 4, 5, 6} OR Experiment Tossing a coin: Sample space S = {Heads, Tails}

Question 4

What are elementary outcomes? Offer examples.

Answer 4

Each single element (outcome) (EO) ,a possible combination NB. Every elementary outcome is an event but NOT VICE VERSA

Question 5

What is an event? Is there a difference between events and elementary outcomes?

Answer 5

Event = set of one or more EO’s for a random experiment = subset of sample space NB. Every elementary outcome is an event but NOT VICE VERSA

Question 6

What are disjoint events? Offer examples of disjoint and non-disjoint events.

Answer 6

Two events (A and B) are disjoint If they have no EO’S in common (A and B) = 0 EXAMPLES _Tossing a coin and getting a heads and a tails at the same time is impossible. _You can’t take the bus and the car to work at the same time. _You can’t get a pay raise and a pay decrease at the same time

Question 7

What is a probability model?

Answer 7

A mathematical description of a random phenomenon consisting of two parts: a sample space (S) and a way of assigning probabilities to events.

Question 8

What is a probability, and which values can probabilities have?

Answer 8

The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is a long-term relative frequency. P = number of outcomes event A / total number of outcomes VALUES = 0 to 1

Question 9

What is the complement rule?

Answer 9

The complement of any event A is the event that A does not occur, written as a^c . Complements are “opposites” (ex. “rain” or “not rain”) P (A^c) = 1 - P(A)

Question 10

What is a uniform probability model?

Answer 10

a model in which every outcome has equal probability. P (A) = number of eo’s in A / number of eo’s in Event

Question 11

How do you determine the probability of an event using a uniform probability model?

Answer 11

P (A) = number of eo’s in A / number of eo’s in Event It is a model in which every outcome has equal probability.

Question 12

How are uniform probability models related to simple random sampling, or SRS?

Answer 12

ex. SRS of size n=3 students , (3 randomly chosen kids) each sample of size n=3 is EO and each EO has the same probability of occurring

Question 13

What is the addition rule and when should it be applied?

Answer 13

General Addition Rule for UNIONS of two events For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) … for disjoint events is =P (A) + P (B)

Question 14

What is the multiplication rule and when should it be applied?

Answer 14

The multiplication rule for independent events says that if A and B are independent, then P(A and B) = P(A) × P(B).  Two events A and B are independent if P(B|A) = P(B) .  Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs.  If A and B are independent, then P(A and B) = P(A) × P(B)  No, disjoint events cannot be independent. If A and B are disjoint, then the fact that A occurs tells us that B cannot occur.  If we know they are disjoint, then if event A happened, we know that event B did not happen, therefore, knowledge of event A affected our knowledge of event B, making them dependent events.

Question 15

What form does the addition rule take on in the case of disjoint events?

Answer 15

 If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then P(one or more of A, B, C) = P(A) + P(B) + P(C).

Question 16

What form does the multiplication rule take on in the special case of independent events?

Answer 16

 Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs.  If A and B are independent, then P(A and B) = P(A) × P(B). How is the general multiplication rule different than the multiplication rule for independent events?  If A and B are independent, then P(B|A) = P(B) so the two rules are the same. If A and B are not independent, the rule must be adjusted for this condition.

Question 17

What is a conditional probability?

Answer 17

The conditional probability, P(B|A), is the probability that event B occurs given that event A has already occurred.

Question 18

How do you use a contingency table to calculate conditional and unconditional probabilities?

Answer 18

unconditional (asolute) probability is a number (inside or total row, columns) devided by the total conditional probability is an inside number in the contingency table divided by the total row/column

Question 19

When are two events (A and B) statistically independent?

Answer 19

 Two events are said to be independent if the outcome of one of them does not influence the other. For example, in sporting events, the outcomes of different games are usually considered independent even though that may not be true in a completely strict and literal sense.  The multiplication rule for independent events says that if A and B are independent, then P(A and B) = P(A) × P(B).  Two events A and B are independent if P(B|A) = P(B) .

Question 20

How can a tree diagram help us to apply the laws of addition and multiplication in an orderly fashion?

Answer 20

A tree diagram is simply a way of representing a sequence of events. Tree diagrams are particularly useful in probability since they record all possible outcomes in a clear and uncomplicated manner.

Question 21

What do we mean by the term random variable?

Answer 21

it is a variable whose possible values are numerical outcomes of a random phenomenon

Question 22

Random variables are distributed according to a specific probability distribution. What does this probability distribution express?

Answer 22

What is probability distribution of random variable X? relative frequency of each of the possible values of X if you were to repeat random experiment an infinite number of times

Question 23

What do we mean by the mean or expected value of a random variable?

Answer 23

Expected value (μX) of X = mean value of random variable X if random experiment were repeated infinite number of times

Question 24

Why is the mean or expected value a weighted average?

Answer 24

weighted average, i.e., each value of X weighted by corresponding probability of occurrence = expected value

Question 25

What do we mean by the standard deviation of a random variable?

Answer 25

used to characterize to what extent the values of the random variable X may differ across repetitions of the random experiment

Question 26

What is the difference between discrete and continuous random variables?

Answer 26

_Discrete variables are countable in a finite amount of time. For example, you can count the change in your pocket. You can count the money in your bank account. You could also count the amount of money in everyone’s bank account. It might take you a long time to count that last item, but the point is — it’s still countable. _Continuous Variables would (literally) take forever to count. In fact, you would get to “forever” and never finish counting them. For example, take age. You can’t count “age”. Why not? Because it would literally take forever. For example, you could be: 25 years, 10 months, 2 days, 5 hours, 4 seconds, 4 milliseconds, 8 nanoseconds, 99 picosends…and so on. You could turn age into a discrete variable and then you could count it. For example: A person’s age in years. A baby’s age in months.

Question 27

What is the Law of Large Numbers?

Answer 27

 In statistical terms, a rule that assumes that as the number of samples increases, the average of these samples is likely to reach the mean of the whole population, i.e., as the number of independent trials increases, the probability based on our observations will get closer and close to the theoretical probability of the event.

Question 28

P (B|A)

Answer 28

probability of B given A - > P (B|A)

Question 29

P (A and B)

Answer 29

probability that both events occur

Question 30

P (A or B)

Answer 30

probability that A or B occur