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Probability

See attached and solve for 1, 3, 5 and 9 only. Give step by step solution with every minor detail.

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Fundamentals in a nutshell

What do you understand by "probability" of an event

If to an action a number of alternative outcomes are possible, then it is not possible to predict the actual outcome in advance with certainty. Only what can be predicted is the possibility (or uncertainty) associated with each possible outcome. The mathematical term for this possibility expressed numerically is probability.

Example : When we throw a die (it is assumed the die is well balanced) we cannot predict the number it will throw up ( 1,2,3,4,5 or 6). However, we can say that the possibility of any one of these numbers being thrown up is equal or the probability of say number 1 being thrown up is 1/6 [P(1) = 1/6].
Similarly, when a coin is tossed, the possibility of head or tail coming up is equal. Or in other words we can say that the probability of head coming up (or tail coming up) is ½.

What is a probability (or probability density) function

The probabilities associated with different possible outcomes can be expressed. For example, in case of a die we can state as follows :

P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6

We can express the above six equations in a single equation :

F(n) = 1/6 for n=1,2,3,4,5,6 } ............(1)
= 0 for any other integer }

Function F(n) is known as a probability function or probability density function (pdf).

From pdf we can also determine the probability of say the die throwing up either 2 or 6. P(2 or 6) = F(2) + F(6) = 1/6 + 1/6 = 2/6
Because the die has to throw up one of the six numbers, therefore P(1,2,3,4,5 or 6) = 6 x 1/6 = 1 .............(2)

Above is an example of discrete probability density function that is the function can take only specified values (integers 1-6). However, the pdf can be a continuously varying function also. For example, if we want to ...

Solution Summary

Step-by-step solution provided on the given probability problems, first outlining the fundamentals.

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