Fill In the Blanks

**DINOCALC09** · October 17th 2010, 10:13 AM

In one toss of a fair coin, we expect ___0.5_____ heads, give or take _________ heads or so.

In n=100 tosses, we expect ___50______ heads, give or take _________ heads or so.

In n=400 tosses, we expect ___200_____ heads, give or take _________ heads or so.

Note that the standard deviation is proportional to the square root of the number of tosses.

I think I got the first fill-in-the-blanks right, but not sure what the 2nd blanks are about. I'm guessing standard deviation. However, I am confused when they define SD as proportional to the sq rt of the number of tosses. Does that mean, sqrt(1) = 1, and sqrt(100) = 10, and sqrt(400) = 20.

**dexteronline** · October 17th 2010, 01:17 PM

I tried the following solution, not sure about the accuracy of the results though

Coin Toss

Number----------- Probability
of Heads--------- of Outcome
x --------------- P(x)
0 --------------- 1/2 = 0.5
1 --------------- 1/2 = 0.5
Total------------ 2/2 = 1

Mean of a probability distribution

$\mu = \sum [xP(x)]$
$\mu = 0(0.5)+1(0.5)$
$\mu = 0.5$

Variance
$\sigma^2 = \sum [(x-\mu)^2P(x)]$
$\sigma^2 = [(0-0.5)^2 (0.5) + (1-0.5)^2 (0.5)]$
$\sigma^2 = [(-0.5)^2 (0.5) + (0.5)^2 (0.5)]$
$\sigma^2 = [(0.25)(0.5) + (0.25)(0.5)]$
$\sigma^2 = [(0.125) + (0.125)]$
$\sigma^2 = 0.25$

Standard Deviation
$\sigma = 0.5$

In one toss of a fair coin, we expect ___0.5_____ heads, give or take ___0.5____ heads or so.

In n=100 tosses, we expect ___50______ heads, give or take ___5___ heads or so

In n=400 tosses, we expect ___200_____ heads, give or take ___10____ heads or so.

**DINOCALC09** · October 17th 2010, 01:40 PM

For SD, you can also use sqrt[(n)(pi)(1-pi)]. I agree with your answers.

Originally Posted by dexteronline

I tried the following solution, not sure about the accuracy of the results though

Coin Toss

Number----------- Probability
of Heads--------- of Outcome
x --------------- P(x)
0 --------------- 1/2 = 0.5
1 --------------- 1/2 = 0.5
Total------------ 2/2 = 1

Mean of a probability distribution

$\mu = \sum [xP(x)]$
$\mu = 0(0.5)+1(0.5)$
$\mu = 0.5$

Variance
$\sigma^2 = \sum [(x-\mu)^2P(x)]$
$\sigma^2 = [(0-0.5)^2 (0.5) + (1-0.5)^2 (0.5)]$
$\sigma^2 = [(-0.5)^2 (0.5) + (0.5)^2 (0.5)]$
$\sigma^2 = [(0.25)(0.5) + (0.25)(0.5)]$
$\sigma^2 = [(0.125) + (0.125)]$
$\sigma^2 = 0.25$

Standard Deviation
$\sigma = 0.5$

In one toss of a fair coin, we expect ___0.5_____ heads, give or take ___0.5____ heads or so.

In n=100 tosses, we expect ___50______ heads, give or take ___5___ heads or so

In n=400 tosses, we expect ___200_____ heads, give or take ___10____ heads or so.

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