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

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 Click here to see ALL problems on Probability-and-statistics Question 630051: A fair coin tossed 64 times. Find the probability of getting 32 to 40 heads inclusiveFound 3 solutions by ewatrrr, Theo, AnlytcPhil:Answer by ewatrrr(10682)   (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 = and SD = *Note: P(x is 32 to 40) = P(z = 2) - P(z = 0) = .9773 - .5 = .4773 ```Answer by Theo(3464)   (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(1278)   (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 = = = 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 = = = -.125 round to hundredths -.13 z = = = 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```