Question

The number of heads (H) reported by all of you is tabulated below, w here the first column is the number of heads and the second column is the number of times it occurred.

0 0

1 0

2 0

3 2

4 6

5 6

6 9

7 3

8 0

9 1

10 0

The number of tails (T) is simply given by H=10-T. The results of the coin flip can be expresses mathematically in a simple way by assigning the value 1 the n a heads occurs and 0 when a tails occurs. You could choose other numerical assignents, but this one is convenient because the total numerical value is the number of heads obtained in each coin flip experiment.

e) The width of the distribution is given byσ=√Npq. What is it for ourexperiment?

f) Plot the probabilities on the plot you did for part a) usinga suitable over-all normalizing factor to account for the number of people doing the coin flipexperiment.You can find discussions of the binomial function in the manual, Yan’s lectures, and on theweb. If your calculator does not do factorials, you can find web-based ones readily. Showyour calculations to get full credit.

(Note - part a -  Plot the results of this experiment, using H on the x-axis and indicating the expected error (√H)) on each measurement.

g) What fraction of our results for the number of times we observed a numberof heads agrees with the binomial distribution (after you normalize is to ourexperiment) within our estimated errors? We will show laterthat about 2/3 ofthem should.

Answer 1

(a) By assuming each coin as unbiased, then p=pr(head in one coin) = 0.5, q = 1-p =1-0.5 = 0.5 then standard deviation sigma = Sqrt(Npq) = Sqrt(10*0.5*0.5)= 1.5811

Using binomial distribution p value is estimated as

Np = mean of the given experimental frequency distribution

10p = 5.33333

p= 0.5333

q=1-p= 1-0.5333= 0.4667

Therefore, the standard deviation sigma from the experimental data is

sigma = Sqrt(Npq) = Sqrt(10*0.5333*0.4667)=1.5776

x	f	Binomial P	Ef	Rounded Ef	error	fx
0	0	0.00049	0.013226	0	0	0
1	0	0.005598	0.151156	0	0	0
2	0	0.028792	0.777374	1	-1	0
3	2	0.087746	2.369136	2	0	6
4	6	0.175491	4.738266	5	1	24
5	6	0.240674	6.498185	7	-1	30
6	9	0.229213	6.188739	6	3	54
7	3	0.14969	4.04162	4	-1	21
8	0	0.064153	1.732121	2	-2	0
9	1	0.016293	0.439903	0	1	9
10	0	0.001862	0.050275	0	0	0
Totals	27	1		27	0	144
mean=5.3333
p=0.533333

(f) The probability plot is

(g)

If we allow the acceptable error in between(-sigma, +sigma) =(-1.5776, +1.5776 )for 2/3 rd cases,

For the given experimental data, we obtain 19(total of bolded expected frequencies 0+0+0+1+2+5+7+4+0+0=19) times out of 27 times the number of heads agrees with the binomial distribution. The fraction is 19/27 = 0.7(approximately), also 2/3 = 0.67 = 07(approximately)