If you flip 10 pennies at once, what are the chances
3 will come up heads and 7 will come up tails? Answer
is 15/128 – but how do you get that answer?
Suppose we number the pennies 1 thru 10.
Let's look at a sample successful case.
penny # 1 2 3 4 5 6 7 8 9 10
T T H T T H T H T T
Let's count all the successful ways:
Of the ten numbered pennies we can choose the pennies
which are to come up heads any of the combinations of
10 things taken 3 at a time, that is
10C3 or C(10,3) or 10!/(3!7!) or 120 way
Now let's count all the possible ways, successful or
unsuccessful:
Each penny may come up either of 2 ways, so that's a
total of 2×2×2×2×2×2×2×2×2×2 or 1024 ways
So the probability is 120/1024 = 15/128.
Another way:
This is a binomial probability of 3 successes out of
10 independent trials, with probability of 1/2 on each
trial.
The formula for x successes out of n independent trials,
with probability of success p is given by the formula:
n!/[x!(n-x)!]·px(1-p)n-x
This problem has n = 10, x = 3, p = 1/5
10!/[3!(10-3)!]·(1/2)x(1-p)n-x = .1171875 = 15/128
Another way
On your TI-83 or better calculator:
Press
CLEAR, 2nd, VARS (to get DISTR menu)
scroll down to binompdf(
after binomcdf(, type 10,1/2,3)
so that you see this on the screen
binompdf(10,1/2,3)
press ENTER
read .1171875, which is the decimal value
of the answer
then press
MATH ENTER ENTER
read 15/128, the fraction answer.
Edwin
