Suppose S = {p, q, r, s, t, u} and A = {p, q, s, t} and B = {r, s, t, u} are events.

x p q r s t u

p(x) 0.15 0.25 0.2 0.15 0.1

(a) Determine what must be p(s).
(b) Find p(A), p(B) and p(A∩B).
(c) Determine whether A and B are independent. Explain.
(d) Arer A and B mutually exclusive? Explain.
(e) Does this table represent a probability istribution of any random variable? Explain.

A person stands at the center of a turntable, holding his arms extended horizontally with a 1.5 kg dumbbell in each hand. He is set rotation about a vertical axis, making one revolution in 2 seconds. Find the new angular velocity if he pulls the dumbbells in to his middle. His moment of inertia (without dumbbells) is 3 kgm2 when his arms are out stretched, dropping to 2.2 kgm2 when his hands are at his middle. The dumbbells are 1 m from the axis initially and 0.2 m from it when moved to his middle.

An enzymatic reaction takes place in a 10 ml solution that has a total citrate (plus citric acid) concentration of 120 mM and an initial pH of 7.00. During the reaction 0.2 millimoles of H+ are produced. The relevant pKa of citric acid is 6.40. A) Calculate the final pH of the solution. B) What would the final pH of the solution be if the citrate were absent from the solution and no other buffering agents were present (assuming a starting pH of 7.00 again, basically, what would be the pH is you added the same amount of acid to that amount of water)?

An investment bank has been asked to underwrite an issue of 10 million shares by a company.

The bank is trying to decide between a best-efforts deal where it charges a fee of $0.2 for each share sold and a firm-commitment deal where it buys the shares for $10 per share.

For the latter deal, the bank considers that the selling price per share is either $10.8 or $9.8.

What are the break-even probabilities of the two selling prices so that the bank is indifferent to the two deals (Assume that all 10 million shares are sold out)?

Question 01: Al Marai offers the customer who buys a 1 kg bottle of milk two free gifts, chosen from the following a chocolate bar, a small bottle of milk, a glass cup, if you purchased milk, what is the probability you randomly select a chocolate bar and a glass cup in that order?

a. 0.00

b. 1.00

c.0.05

d. 0.2

.

Question 02: A box of 24 types of chocolate bars that are made in UAE. Three bars are to be chosen randomly for testing.

how many different combinations of three bars could be selected

A stock's returns have the following distribution:

Calculate the stock's expected return. Round your answer to two decimal places.

Calculate the stock's standard deviation. Do not round intermediate calculations. Round your answer to two decimal places.

Calculate the stock's coefficient of variation. Round your answer to two decimal places.

Question 4

An interesting topic in fluid mechanics is Bernoulli's equation, which relates pressures and velocities of a fluid with the density of rho at different heights, eq. 13.12. It has applications in the airspace. It can be used to measure flow speed in a pipe with a device named the Venturi meter (see figure 13.34). If the cross sections are 21 cm2 and 13 cm2, and the height difference resulted from the pressure difference is 0.2 m, calculate the speed of the flow in the pipe, v1 . Give your answer to one decimal point. Your Answer:Question 4 options:

Answer units

A company makes TV advertisements to promote its products. Based on the survey results it is known: The chance of someone buying the product is 0.2, the chance someone sees the ad is 0.4, and the chance of someone buying the product and seeing the ad is 0.12.
a. what is the chance someone does not buy the product and see the ad b. what is the chances of someone buying a product if it is known he saw an ad
c. what is the chance for someone to buy the product if it is known he did not see the ad
d. Is the occurrence of a product purchase independent of the incident seeing the ad

Answer using R Studio

1. Here we consider the amount of data needed to perform hypothesis testing.
   1. Suppose we are testing a coin using observations of tosses. We wish to test H₀: p = 0.5 against an alternative of H_A : p = 0.6 (in this question use one-sided tests only). How many tosses are needed to guarantee a size Æ∑ 0.05 and Ø∑ 0.2?
   2. Now generalize to consider H_A : p = 0.5+delta. Choose sensible values for delta and quantify the number of observations needed as a function of delta

The amounts a soft drink machine is designed to dispense for each drink are normally distributed, with a mean of 11.7 fluid ounces and a standard deviation of 0.2 fluid ounce. A drink is randomly selected.

(a) Find the probability that the drink is less than 11.6 fluid ounces.

(b) Find the probability that the drink is between 11.5 and 11.6 fluid ounces.

(c) Find the probability that the drink is more than 12 fluid ounces. Can this be considered an unusual event? Explain your reasoning.

Is a drink containing more than 12 fluid ounces an unusual event?