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. The formula specified is actually called the multiplication rule and has important applications. 

P(A)=0.40  P(U/A)=0.10  P(U/A).P(A)=0.04

P(B)=0.35  P(U/B)=0.20  P(U/B).P(B)=0.07

P(C)=0.25  P(U/C)=0.05  P(U/C).P(C)=0.0125

P(U)= 0,04+0,07+0,0125=0,12

P(B/U)=0,07/0,12=0,58

The standard deviation of the continuous random variable X;

In statistical notation random variables represented by capital letters such as X,Y and so on. So number III is false. Number I and II are true. The correct answer is B. 

f(x) = 1 / (b - a) ; E(X) = (a + b) / 2 = (-5 + b) / 2 = 10 ; b = 25. pg. 186. Correct answer is E.

A random experiment is any process that leads to two or more possible outcomes, without knowing
exactly which outcome will occur. The set of all possible outcomes of a random experiment is called the sample space. Each possible outcome of a random experiment is called an elementary outcome. An event occurs if the random experiment results in one of the basic outcomes of that event. A is the correct answer.

I. P(A+B) = P(A?B)P(B) this formula is called the multiplication rule. (True)

II.  A and B are statistically independent if and only if P(A+B) = P(A)P(B). (True)

III. For A and B sets independence can also be denoted by P(A?B). (False, P(A?B) denotes conditional probability for sets A and B.)

The answer is C. 

I. 0 ? F (x) ? 1 (True)

II.  If x₁ ? x₂ then F (x₁) ? F (x₂) (True)

III.  If x₁ = x₂ then F (x₁) = F (x₂) (False,  If x₁ ? x₂ then F (x₁) ? F (x₂))

The answer is C.

1. P(Z < (-a)) = P(Z > a) = 0.5 – 0.195 = 0.305 . pg. 193. Correct answer is C.

Empirical Probability of an Event = The number of times the event happens/Total number of observations

P (Number 2)= 156 / 1000 =0,156

Note that this event corresponds to (O+Y –) , ( – O+Y – ) , ( – O+Y – ), and that (O+Y –) , ( – O+Y – ) , ( – O+Y – ) , (O+Y ) = S.
Therefore, P((O+Y – ) , ( – O+Y –) , ( –O+Y–)) = 1 – P(O+Y– ) = 1 – 0.03 = 0.97 The correct answer is B.

A major difference between continuous and discrete random variables is the former takes on uncountable and infinite number of possible outcomes in a given interval. Hence the range of continuous random variable X comprises all real numbers in an interval. 

What are the “big data” sets today and where do they come from? These are mostly found in the following areas:

• Commerce, especially online electronic commerce
• Finance, for example share prices on stock markets, all managed electronically
• Insurance, all the premiums, incidents, actuarial transactions in an insurance company
• Biomedicine, especially in genetics, where information is literally exploding as gene-sequencing reveals and codes the total genetic profile of a person
• Transport, for example in the airline industry, all the flights, all the passengers
• Climate data, measurements from tens of thousands of weather stations across the world

P(Z > a) = 0.5 – 0.17 = 0.33. pg. 193. Correct answer is C.

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 judgment comes in to play to assign the probability. Thi­s approach is not favorable method to assign probability, but sometimes i­f there’s no previous knowledge on the subject then the researcher may assign a subjective probability as a starting point.