SOLUTION: A fair coin tossed 64 times. Find the probability of getting 32 to 40 heads inclusive

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Question 630051: A fair coin tossed 64 times. Find the probability of getting 32 to 40 heads inclusive
Found 3 solutions by ewatrrr, Theo, AnlytcPhil:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi,

A fair coin tossed 64 times. p= 1/2 and q = 1/2,  mean =64%2A%281%2F2%29=+32  and SD = sqrt%2864%2A%281%2F2%29%2A%281%2F2%29%29+=+4

*Note: z+=+%28x+-+mu%29%2F%28sigma%29

 P(x is 32 to 40) = P(z = 2) - P(z = 0) =  .9773 - .5 = .4773

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
this is going to be equal to the probability of getting:
32 heads + probability of getting
33 heads + probability of getting
34 heads + probability of getting
35 heads + probability of getting
36 heads + probability of getting
37 heads + probability of getting
38 heads + probability of getting
39 heads + probability of getting
40 heads.
since this is a binomial type of probability problem, the formula would be:
probability of getting x heads = C(n,x) * p^x * q^(n-x)
n is equal to the number of tosses which is equal to 64
x is equal to the desired number of heads.
p is equal to the probability of getting a heads which is equal to .5
q is equal to the probability of not getting a heads which is also equal to .5 since the total probability of getting a heads and not getting a heads has to be equal to 1.
C(n,x) is the combination formula of n! / (x! * (n-x)!)
i cheated by using excel to calculatoe the probbabilities and this is what i got.
the sum of all the probabilities is supposed to equal 1 which it does so i'm reasonably sure that i did it right.
x	C(n,x)	        p^x	        q^(n-x)	        p(n,x)
0	1	        1	        5.42101E-20	5.42101E-20
1	64	        0.5	        1.0842E-19	3.46945E-18
2	2016	        0.25	        2.1684E-19	1.09288E-16
3	41664	        0.125	        4.33681E-19	2.25861E-15
4	635376	        0.0625	        8.67362E-19	3.44438E-14
5	7624512	        0.03125	        1.73472E-18	4.13326E-13
6	74974368	0.015625	3.46945E-18	4.06437E-12
7	621216192	0.0078125	6.93889E-18	3.36762E-11
8	4426165368	0.00390625	1.38778E-17	2.39943E-10
9	27540584512	0.001953125	2.77556E-17	1.49298E-09
10	1.51473E+11	0.000976563	5.55112E-17	8.21138E-09
11	7.43596E+11	0.000488281	1.11022E-16	4.03104E-08
12	3.28421E+12	0.000244141	2.22045E-16	1.78038E-07
13	1.31369E+13	0.00012207	4.44089E-16	7.12151E-07
14	4.78557E+13	6.10352E-05	8.88178E-16	2.59426E-06
15	1.59519E+14	3.05176E-05	1.77636E-15	8.64754E-06
16	4.88527E+14	1.52588E-05	3.55271E-15	2.64831E-05
17	1.37937E+15	7.62939E-06	7.10543E-15	7.47758E-05
18	3.60169E+15	3.8147E-06	1.42109E-14	0.000195248
19	8.71988E+15	1.90735E-06	2.84217E-14	0.000472706
20	1.96197E+16	9.53674E-07	5.68434E-14	0.001063587
21	4.1108E+16	4.76837E-07	1.13687E-13	0.002228469
22	8.03474E+16	2.38419E-07	2.27374E-13	0.004355644
23	1.46721E+17	1.19209E-07	4.54747E-13	0.007953785
24	2.50649E+17	5.96046E-08	9.09495E-13	0.013587715
25	4.01039E+17	2.98023E-08	1.81899E-12	0.021740344
26	6.01558E+17	1.49012E-08	3.63798E-12	0.032610517
27	8.46637E+17	7.45058E-09	7.27596E-12	0.045896283
28	1.11877E+18	3.72529E-09	1.45519E-11	0.060648659
29	1.38882E+18	1.86265E-09	2.91038E-11	0.075287991
30	1.62029E+18	9.31323E-10	5.82077E-11	0.087835989
31	1.77709E+18	4.65661E-10	1.16415E-10	0.096336246
-------------------------------------------------------------------
32	1.83262E+18	2.32831E-10	2.32831E-10	0.099346754
33	1.77709E+18	1.16415E-10	4.65661E-10	0.096336246
34	1.62029E+18	5.82077E-11	9.31323E-10	0.087835989
35	1.38882E+18	2.91038E-11	1.86265E-09	0.075287991
36	1.11877E+18	1.45519E-11	3.72529E-09	0.060648659
37	8.46637E+17	7.27596E-12	7.45058E-09	0.045896283
38	6.01558E+17	3.63798E-12	1.49012E-08	0.032610517
39	4.01039E+17	1.81899E-12	2.98023E-08	0.021740344
40	2.50649E+17	9.09495E-13	5.96046E-08	0.013587715
-------------------------------------------------------------------
41	1.46721E+17	4.54747E-13	1.19209E-07	0.007953785
42	8.03474E+16	2.27374E-13	2.38419E-07	0.004355644
43	4.1108E+16	1.13687E-13	4.76837E-07	0.002228469
44	1.96197E+16	5.68434E-14	9.53674E-07	0.001063587
45	8.71988E+15	2.84217E-14	1.90735E-06	0.000472706
46	3.60169E+15	1.42109E-14	3.8147E-06	0.000195248
47	1.37937E+15	7.10543E-15	7.62939E-06	7.47758E-05
48	4.88527E+14	3.55271E-15	1.52588E-05	2.64831E-05
49	1.59519E+14	1.77636E-15	3.05176E-05	8.64754E-06
50	4.78557E+13	8.88178E-16	6.10352E-05	2.59426E-06
51	1.31369E+13	4.44089E-16	0.00012207	7.12151E-07
52	3.28421E+12	2.22045E-16	0.000244141	1.78038E-07
53	7.43596E+11	1.11022E-16	0.000488281	4.03104E-08
54	1.51473E+11	5.55112E-17	0.000976563	8.21138E-09
55	27540584512	2.77556E-17	0.001953125	1.49298E-09
56	4426165368	1.38778E-17	0.00390625	2.39943E-10
57	621216192	6.93889E-18	0.0078125	3.36762E-11
58	74974368	3.46945E-18	0.015625	4.06437E-12
59	7624512	        1.73472E-18	0.03125	        4.13326E-13
60	7624512	        1.73472E-18	0.03125	        4.13326E-13
61	41664	        4.33681E-19	0.125	        2.25861E-15
62	2016	        2.1684E-19	0.25	        1.09288E-16
-------------------------------------------------------------------
63	64	        1.0842E-19	0.5	        3.46945E-18
-------------------------------------------------------------------
64	1	        5.42101E-20	1	        1.76183E-18

E-k means the number before the E * 10^-k
k represents the number following the - sign.
example
the probability of getting 63 heads is equal to:
C(64,63) * (.5)^63 * (.5)^(1) which is equal to:
C(64,63) * (.5)^63 * (.5)^1 which is equal to 3.46945 * 10^-18
if you look at the entry for x = 63, you'll see that the probability is 3.46945E-18 which is the same as 3.46945 * 10^-18
the answer to your question is that the probability of getting 32 to 40 heads inclusive is equal to 0.533290497 which is the sum of the probabilities of getting exactly 32 heads plus exactly 33 heads ..... plus exactly 40 heads.

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
A fair coin tossed 64 times. Find the probability of getting 32 to 40 heads inclusive

The first tutor forgot to subtract .5 from the lower bound and to add .5
to the upper bound when using the normal to approximate the binomial.
The second tutor got it correct using Excel to do the calculations.  In fact
it is probably more accurate than below, but I think your teacher intended you
to use the following method.

You're probably supposed to use the normal approximation, as the first tutor
assumed but she forgot about the ".5". 

Calculate the mean = n×p = 64×.5 = 32

Standard deviation = sqrt%28n%2Ap%2A%281-p%29%29 = sqrt%2864%2A.5%2A.5%29 = 4

Then calculate the z-scores for x = 31.5 and 40.5.  (we subtract .5 from
the lower bound 32 and add .5 to the upper bound 40).

z = %28x-mu%29%2Fsigma = %2831.5-32%29%2F4 = -.125 round to hundredths -.13 

z = %28x-mu%29%2Fsigma = %2840.5-32%29%2F4 = 2.125 round to hundredths 2.13

We look those z-values up in the normal table. Depending on which kind of
normal table you have, you do one of these: 

  subtract .9834 - .4483 = .5351  if your table has negative values of z.      

or you add .0517 + .4834 = .5351  if your table has only positive z values.

Either way the answer is .5351
 
Edwin