Expected Value Of The Sum Of The Draws?
Problem 1
One hundred draws volition be made at random with replacement from the box
(a) How small can the sum of the draws be? How big?
(b) The sum is between 650 and 750 with a take a chance of about
1$\% \quad 10 \% \quad 50 \% \quad 90 \% \quad 99 \%$
Bryan L.
Numerade Educator
Problem 2
A gambler plays roulette 100 times, betting a dollar on a column each time The bet pays ii to $ane,$ and there are 12 chances in 38 to win. Fill in the blanks prove work.
(a) In 100 plays, the gambler's net gain will be effectually $\$$ accept $\$$
(b) In 100 plays, the gambler should win $\longrightarrow$ times, give or accept (c) How does the column bet compare with betting on a single number at Keno (example one on p. 289)?
Heena H.
Numerade Educator
Problem 3
Match the lists with the SDs. Explain your reasoning
$\begin{array}{ll}{\text { (a) } 1,-2,-2} & {\text { (i) } \sqrt{ane / iii \times 2 / 3}} \\ {\text { (b) } xv,15,sixteen} & {\text { (ii) } ii \times \sqrt{1 / 3 \times two / 3}} \\ {\text { (c) }-1,-ane,-1,1} & {\text { (ii) } 3 \times \sqrt{ane / 3 \times 2 / 3}} \\ {\text { (d) } 0,0,0,1} & {\text { (iv) } \sqrt{one / iv \times iii / iv}} \\ {\text { (e) } 0,0,2} & {\text { (v) } 2 \times \sqrt{i / 4 \times 3 / iv}}\cease{assortment}$
Bryan Fifty.
Numerade Educator
Trouble four
A big grouping of people gather. Each i rolls a dice 180 times, accounts the number of $["$ 'due south. About what pct of these people should counts in the range 15 to 45$?$
Heena H.
Numerade Educator
Trouble v
A die will be thrown some number of times, and the object is to approximate the total number of spots. At that place is a one-dollar penalty for each spot that the guess is off. For example, if you approximate 200 and the total is $215,$ you lose $\$ 15 .$ Which practise you adopt: 50 throws, or 100$?$ Explicate.
Bryan Fifty.
Numerade Educator
Trouble half-dozen
One hundred draws are fabricated at random with replacement from the box
$\lfloor\foursquare[1][2]$ The draws come out as follows: $45[ane]^{\prime} southward, 23[2]^{\prime number} s, 32[3]^{\prime} s$ For each number beneath, find the phrase which describes it.
Heena H.
Numerade Educator
Problem vii
One hundred draws are made at random with replacement from the box $\lfloor[1][2][3][iv][5][6]$
(a) If the sum of the draws is $321,$ what is their average?
(b) If the boilerplate of the draws is $iii.78,$ what is the sum?
(c) Estimate the chance that the average of the draws is between 3 and four.
Bryan L.
Numerade Educator
Problem 8
A coin is tossed 100 times.
(a) The difference "number of heads - number of tails" is like the sum of 100 draws from 1 of the following boxes. Which 1, and why?
(b) Find the expected value and standard fault for the difference.
Heena H.
Numerade Educator
Problem 9
A gambler plays roulette $1,000$ times. There are ii possibilities:
(i) Betting $\$ 1$ on a column each time.
(ii) Betting $\$ one$ on a number each time.
A column pays 2 to $1,$ and there 12 chances in 38 to win; a number pays
35 to $1,$ and there is one risk in 38 to win. Truthful or false and explain:
(a) The chance of coming out ahead is the aforementioned with (i) and (ii).
(b) The chance of winning more than $\$ 100$ is bigger with (ii).
(c) The take a chance of losing more than than $\$ 100$ is bigger with (two).
Problem x
A box contains numbered tickets. Draws are madom with replacement from the box. Beneath are 3 statements near this particular box; (i) and (two) are true. Is (three) true or false? Explain. (i) For a sure number of draws, the expected value for the sum of the draws equals $400 .$
(ii) For that same number of draws, at that place is well-nigh a 75$\%$ risk that the sum will be between 350 and $450 .$
(3) For twice that number of draws, there is about a 75$\%$ adventure that the sum will be between 700 and 900 .
Heena H.
Numerade Educator
Problem 11
1 hundred draws are fabricated at random with replacement from the box
The sum of the positive numbers will be around
Bryan 50.
Numerade Educator
Problem 12
1 hundred draws are made at random with replacement from the box
Problem 12
I hundred draws are made at random with replacement from the box
The sum of the draws is $431 .$ The expected value for the sum of the draws is $\longrightarrow$ , the observed value is
(b) The sum of the draws is $386 .$ The expected value for the sum of the draws is $\longrightarrow$ , the observed value is the chance fault is $\quad$ and the standard mistake is
Heena H.
Numerade Educator
Problem thirteen
A letter is fatigued $1,000$ times, at random, from the give-and-take A R A B I A. There are two offers.
(A) Y'all win a dollar if the number of A's among the draws is 10 or more than above the expected number.
(B) You lot win a dollar if the number of B's among the draws is 10 or more above the expected number.
Bryan Fifty.
Numerade Educator
Problem 14
In roulette, once in a while, someone volition bet $\$ 1$ on red; and, at the same time, someone else will bet $\$ one$ on black (p. $282 ) .$ Suppose this pair of bets is made 100 times in the course of an evening.
(a) The house will make money on of the 100 pairs of bets, requite or take $\longrightarrow$ or so.
(b) The cyberspace gain for the house from the 100 pairs of bets will be around $\longrightarrow$ give or take $\longrightarrow$ or then.
Heena H.
Numerade Educator
Source: https://www.numerade.com/books/chapter/the-expected-value-and-standard-error/
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