Statistics 1 Dersi 4. Ünite Sorularla Öğrenelim

Define random experiment.

A random experiment is any process that leads to two or more possible outcomes, without knowing exactly which outcome will occur.

Define sample space and elementary outcome with their notation.

The set of all possible outcomes of a random experiment is called the sample space (S). Each possible outcome of a random experiment is called an elementary outcome(E).

What is the condition for two events being mutually exclusive?

If A and B are any two events then they are said to be mutually exclusive if AnB = Ø. Namely, the intersection set of events must be empty set if they are mutually exclusive.

Give an example of mutually exclusive events.

Having a "head" and having a "tail" when throwing a coin are mutually exclusive.

What do we assume in the classical probability approach?

In the classical probability approach to assign a probability to an event, the assumption is that all the outcomes have the same chance of happening.

Which type of propability approaches is based on experiments?

The empirical probability is based on experperiments.

Define and explain subjective probability.

Sometimes it may not possible to observe the outcomes of events; therefore, the researcher may assign a probability to an event. In subjective probability approach, the researcher assigns a suitable value as the probability of the event. Therefore, a personal judgement comes in to play to assign the probability.

What is the difference between permutation and combination?

When counting the possible outcomes of an event, it is sometimes important to distinguish between ordered and unordered arrangements. When order is important, we call the arrangements of a finite number of distinct objects a permutation. When order is not important, the arrangements of r objects from n distinct objects is called a combination.

A group of 3 students will be choosen among 10 students. How many different groups are possible?

Since the order of students is not important we can use combination. Thus:

C(10,3)=10!/7! 3!=7!*8*9*10/7!*3*2=4*3*10=120. So 120 different groups are possible.

Assume the probability of event "Ali finishes his project in 10 days" is 0.4 and the probability of "Aysel finishes her project in 10 days" is 0.7. What is the probability of at least one project finish in 10 days?

In 10 days either 2 projects will finish(P(2)) or 1 project will finish(P(1)) or none (zero) project will finish (P(0)) in 10 days. Thus:

P(0)+P(1)+P(2)=1

We are interested in at least one project to finish. So we have to compute P(1)+P(2)

P(1)+P(2)=1-P(0)

Since P(Ali Finish)=0.4, then P(Ali Doesnt Finish)=1-0.4=0.6

Since P(Aysel Finish)=0.7, then P(Ali Doesnt Finish)=1-0.7=0.3

Thus P(0)=P(Neither Ali nor Aysel Finish)=06*0.3=0.18

Therefore:

P(1)+P(2)=1-P(0)=1-0.18=0.82

Define conditional probability.

Suppose that we know that event B has occurred and we are interested in finding the probability of event A. That is, we are interested in finding the probability of A knowing that event B has occurred. This is denoted by P(A|B) and is defined as:

P(A|B=P(AnB)/P(B)

Define statistical independence.

In general, we say that the events A1, A2, ..., Ak are mutually statistically independent if and only ifP(A1 ?A2 ?... ?Ak) = P(A1)P(A2)...P(Ak)

A dice is rolled. We know that the number on face is greater than 3. What is the probability that it is also smaller than 6?

If the dice is greater than 3, then the event space is

S=(4,5,6)

So, if we want it to be smaller than 6, we want the dice come 4 or 5. Therefore the probability is 2/3.

There are 20 balls in a bag of whom 5 are blue, 8 are red and 7 are pink. One picks a ball from the bag, and he says that it is not pink. What is the probability that it is blue?

The probability of not being pink=13/20

The probability of being blue=5/20

Then probability of being blue, given not being pink=(5/20)/(13/20)=5/13

There are 20 male and 30 female students in a class. 1/4 of male students and 1/3 of female students are wearing glasses. A student is choosen from this class. What is the probability of student being a female, if it is known that student wears glasses?

Since 1/4 of males have glasses, there are 1/4*20=5 males with glasses and 15 without glasses.

Since 1/3 of females have glasses, there are 1/3*30=10 females with glasses and 20 without glasses. Thus we can prepare the following table. From the table it is easily seen that 2/3 of students with glasses are female.(10/15=2/3)

A dice and coin are thrown together. What is the probability of having a head on coin and a number smaller than 3 on dice?

Let P(A) denote having a head and P(B) having a number smaller than 3.

P(A)=1/2

P(B)=1/3 (since we want either 1 or 2 come on dice; 2/6=1/3)

Since these are independent events P(AnB)=P(A)*P(B)=1/2*1/3=1/6

Compute the following probabilities according to given table above about the car models and their colors.

P(Red/Model C)

P(Model C/Red)

P(Red/Model C)=15/50=0.3

P(Model C/Red)=15/40=0.375