Calculate about the probability

1. A fair coin is tossed 10 times. What is the probability that exactly 6 heads will occur.

2. If 3% of the electric bulbs manufactured by a company are defective find the probability

that in a sample of 100 bulbs exactly 5 bulbs are defective.

3. An oil exploration firm is formed with enough capital to finance 10 explorations. The

probability of a particular exploration being successful is 0.1. Find mean and variance of

the number of successful explorations.

4. Emily hits 60% of her free throws in basketball games. She had 25 free throws in last

week's game.

a) What is the expected number and the standard deviation of Emily's hit ?

b) Suppose Emily had 7 free throws in yesterday's game.What is the probability that she

made at least 5 hits?

5. A coin is loaded so that heads has 60% chance of showing up. This coin is tossed 3 times.

a) What are the mean and the standard deviation of the number of heads that turned out?

b) What is the probability that the head turns out at least twice?

c) What is the probability that an odd number of heads turn out in 3 flips?

6. According to the 2009 current Population Survey conducted by the U.S. Census Bureau,

40% of the U.S. population 25 years old and above have completed a bachelor's degree or

more. Given a random sample of 50 people 25 years old or above, what is expected

number of people and the standard deviation of the number of people who have

completed a bachelor's degree.

7. Joe throws a fair die six times and face number 3 appeared twice. It he incredibly lucky or

unusual?

8. If the probability of being a smoker among a group of cases with lung cancer is .6, what's

the probability that in a group of 8 cases you have; (a) less than 2 smokers? (b0 More

than 5? (c) What are the expected value and variance of the number of smokers?

9. The manufacturer of the disk drives in one of the well-known brands of microcomputers

expects 2% of the disk drives to malfunction during the microcomputer's warranty

What is meant by the term nominal design in connection with a radial flow gas turbine rotor? Sketch

the velocity diagrams for a 90 IFR turbine operating at the nominal design point. At entry to a 90

IFR turbine the gas leaves the nozzle vanes at an absolute flow angle, ?2, of 73. The rotor blade tip

speed is 460 m/s and the relative velocity of the gas at rotor exit is twice the relative velocity at rotor

inlet. The rotor mean exit diameter is 45% of the rotor inlet diameter. Determine,

(i) the exit velocity from the rotor;

(ii) the static temperature difference, T2 - T3, of the flow between nozzle exit and rotor exit.

Assume the turbine operates at the nominal design condition and that Cp 1.33 kJ/kg K.

10. The initial design of an IFR turbine is to be based upon Whitfield's procedure for optimum efficiency. The turbine is to be supplied with 2.2 kg/s of air, a stagnation pressure of 250 kPa, a stagnation temperature of 800C, and have an output power of 450 kW. At turbine exit the static

pressure is 105 kPA. Assuming for air that ? 1.33 and R 287 J/kg K, determine the value

of Whitfield's power ratio, S, and the total-to-static efficiency of the turbine.

Q1: (8 point, 2 point each part) For each of the following part calculate the probability. a) X has a normal distribution with mean 65 and standard deviation 5. P(62 7) = H1. Compute a) E(X), b) Variance and c) standard deviation of a binomial distribution of defective bolts in which P=0.05 and n=55. d) use poison approximation to calculate the possibility of 2 bolts are defective. Only c should be uploaded here. upload a, b and d in Dropbox.Exercise 2.37 If X has the binomial distribution with parameters , and p = 1 -q, show that E(X) = np. E(X3 ) - npq + 13 p2. and deduce the variance of X. Exercise 2.38 Show that var(aX + b) = a var(X) for a, be R. Exercise 2.39 Find E(X) and E(X") when X has the Poisson distribution with parameter A, and hence show that the Poisson distribution has variance equal to its mean.Mean, Variance, and Standard Deviation for the Binomial Distribution The mean, variance, and the standard deviation of a variable that has the binomial distribution can be found by using the following formulas. Mean: ut= n. p Variance: o' =n.p.q Standard deviation: G = n. p.q Example # 23: Find the mean, variance, and standard deviation for the number of heads when ten coins are tossed