Alan

Let v be speed in km/hr on outward journey.

Outward time:  10/v hr

Return time: 10/(v - 3) hr

Total time:  10/v + 10/(v - 3) = 7

Can you take it from here?
 

Marvin roll a die. If he obtain a 5, he chooses a banana from box 1 where 4 are green and 6 are yellow. If he obtain a number that is not a 5, he chooses a banana from box 2 where 8 bananas are green and 3 bananas are yellow. What is the probability of obtaining a colored green banana?

P(green banana) = (1/6)*(4/10) + (5/6)*(8/11) 

I'll leave you to simplify the numbers.

I interpreted the question as requiring a different experiment for each event.  However, if you need a single experiment, then how about something like:

Experiment:   

Throw a standard die.

Events:

a)  The probability of getting a number between 1 and 6 inclusive = 100%

b) The probability of getting an even number = 1/2

c) The probability of getting a 1 = 1/6

d) The probability of getting a 7 = 0

If there is some other interpretation of your question you will need to clarify!

The probability of throwing a 1 with a standard six-sided die is 1/6 (which I assume is what part  c) is asking for).

For part a) the probability of throwing a 1 with a six-sided die with 1 on every face is 100%.

etc.

Let x and y be integers. Show that 9x+5y is divisible by 19 if and only if x+9y is divisible by 19.

As follows:

As follows:

P(2≤X≤5) is given by the area under the curve between 2 and 5.  This is:

(1/2)*(4-2)*(0.2-0.05) + (1/2)*(5-4)*(0.15-0.05) + (5-2)*0.05  =  0.35

Write 16 as 2⁴ = (1/2)^-4   so  (1/2)^-4x = (1/2)^(4-x)  

Therefore  -4x = 4 - x  or  3x = -4   so  x = -4/3

Your answers are correct.  I don't think there's a need to express them as fractions as the probabilities that are supplied are not given as fractions.