Question

1. If you are tossing a fair coin 4 times and X - the absolute difference between the number of HEADs and the number of TAILs, write down the pmf table for X. Find its mean and the standard deviation. [51 2. If you are drawing two cards (one-by-one with replacement), without looking, from a well-shuffled deck of 52 cards, what is the total count of the sample space? What is P(both are aces)? What is P(at least one of them is a face card)? What is the conditional probability that both are spades, given that both are black? 3. (a) You run a small business that has 10 employees. During the current flu season, there is a 20% chance that a randomly chosen employee will be sick with flu before the end of the season. The 10 employees sit in 10 separate closed-door offices in a building and have no way of infecting each other. What are the chances that half of your employees will be down with flu before the end of the year? That none of them will? (b) In your home town, the crime rate is very low. There is only a 1 in 200 chance of a house being broken into by a burglar in a one-month period. Among the 1000 households in this town, what are the chances that 10 will be broken into by burglars during the next month? 4. It is known that about 30% of the students who enroll into the Statistics for Everyday Life' course at the University of Akron drop out of that course. If you randomly select 6 students who have enrolled in that course, what are the chances that (a) none of them will drop out? (b) exactly half of them will drop out? (c) all but 2 will drop out? 5. You have a special die with ‘3, marked on three of its sides,"2, marked on two of its sides and 'I' on the remaining side. You roll the die twice (call it a 'trial'). If both rolls produce the same number, call it a 'success'. What are the chances of a success in a single trial? If you make these trials (a trial-two rolls of that die) 5 times, what are the chances that there are (a) more than 1 'success'? (b) at most 2 successes? 6. An urn contains 4 balls, half of which are red and the other half are white. Reaching into this urn and without looking, you grab 2 balls all at once. Let X -the number of red balls among the 2 you grabbed. Let Y 1 if the number of white balls in your sample 2 is no less than the number of red balls (Y-0 otherwise). Write down the pmL table for X and find its mean. Also, write down the pmf table for Y and find its mean. 15) 7. A water source (say, a fresh-water lake, call it A) and a water reservoir (call it B) are connected by a pipeline that has two arms. The upper arm has one valve (call it V1), while the lower arm has two valves (call them V2 and V3) in series connection (i.e. one after another). The 3 valves are operated independently. If on a randomly chosen day

Answer 1

1)

pmf of X is as follows:

P(X=0)=P(2 heads and 2 tails)=⁴C₂ (0.5)²(0.5)² =0.375

P(X=2)=P(3 heads and 1 tails)+P(1 heads and 3 tails)=⁴C₃ (0.5)³(0.5)¹ +⁴C₃ (0.5)³(0.5)¹ =0.25+0.25=0.50

P(X=4)=P(4 heads and 0 tails)+P(0 heads and 4 tails)=⁴C₄ (0.5)⁴(0.5)⁰ +⁴C₀ (0.5)⁰(0.5)⁴=0.0625+0.0625=0.1250

x	f(x)	xP(x)	x2P(x)
0	0.3750	0.000	0.000
2	0.5000	1.000	2.000
4	0.1250	0.500	2.000
	total	1.500	4.000
	E(x) =μ=	ΣxP(x) =	1.5000
	E(x2) =	Σx2P(x) =	4.0000
	Var(x)=σ2 =	E(x2)-(E(x))2=	1.7500
	std deviation=	σ= √σ2=	1.3229

from above mean =E(X)=1.5

and std deviation =1.3229