please explain the multiplication rule as it applies to the Rules of Probability in at least 400 words

Let X be distributed as a geometric with a probability of success of 0.10. Find the probability it takes 10 or more trials to get the first success.

In a normal disruption, what is the probability that a data value will fall above the data value associated with a z-score of 0.15?

If the interference value is 1, then what is the probability of recombination between genes A and B if there is a recombination between genes B and C, where the gene order is A-B-C, the recombination fraction between A and B is 0.05, and recombination fraction between B and C is 0.15?

A. 0

B. 0.20

C. 1

D. 0.15

E. 0.05

Suppose that the probability that a passenger will miss a flight is 0.0928. Airlines do not like flights with empty​ seats, but it is also not desirable to have overbooked flights because passengers must be​ "bumped" from the flight. Suppose that an airplane has a seating capacity of 54 passengers. ​(a) If 56 tickets are​ sold, what is the probability that 55 or 56 passengers show up for the flight resulting in an overbooked​ flight? ​(b) Suppose that 60 tickets are sold. What is the probability that a passenger will have to be​ "bumped"? ​(c) For a plane with seating capacity of 59 ​passengers, how many tickets may be sold to keep the probability of a passenger being​ "bumped" below 5​%?

1. Calculate (by hand) the mean and standard deviation for the binomial distribution with the probability of a success being 0.50 and n = 10. Either show work or explain how your answer was calculated. Use these formulas to do the hand calculations: Mean = np, Standard Deviation =  Comparison:
2. Calculate (by hand) the mean and standard deviation for the binomial distribution with the probability of a success being 0.10 and n = 10. Write a comparison of these statistics to those from question 5 in a short paragraph of several complete sentences. Use these formulas to do the hand calculations: Mean = np, Standard Deviation = Comparison:
3. Calculate (by hand) the mean and standard deviation for the binomial distribution with the probability of a success being 0.90 and n = 10. Write a comparison of these statistics to those from question 6 in a short paragraph of several complete sentences. Use these formulas to do the hand calculations: Mean = np, Standard Deviation =   
   
     

Derive conditional expectation from joint probability density function.

a) A pair of fair dice is thrown. What is the probability of rolling a value between 8 and 11, inclusive? (Write your answer as a decimal rounded to 3 decimal places.)

b) What is the probability of drawing a black face card when a single card is randomly drawn from a standard deck of 52 cards? (Write your answer as a decimal rounded to 3 decimal places.)

Suppose that there are two identical and independent projects, each with a probability of 0.03 of a loss of $8m and a probability of 0.97 of a loss of $2m. Calculate the 96% VaR and expected shortfall for each project considered separately and the two projects combined. Comment on the quality of subadditivity for VaR and expected shortfall based on your results

to find a probability of values in a tail of the normal distribution using a standard normal table