ADDITIONAL PP 9.1
If twelve unbiased coins are flipped once, what is the probability of getting
a.
b.
c.
d.
Exactly 12 heads?
At least 11 heads?
As extreme or more extreme than 11 heads?
If the 12 coins were biased such that the probability of a head with each is 0.40, what is the
probability of getting exactly 12 heads?
SOLUTION is on next page.
SOLUTION
a.
Because there are 12 coins, N = 12. Let P = the probability of getting a head in one flip of
any coin. Since the coins are unbiased, P = Q = 0.50. The number of P events equals 12.
Entering Table B under N = 12, number of P events = 12 and P = 0.50, we find
p(exactly 12 heads) = 0.0002
b.
Given there are 12 coins, N = 12. Let P = the probability of getting a head in one flip of any
coin. Since the coins are unbiased, P = Q = 0.50. Since the problem asks for “at least 11
heads,” the number of P events is 11 or 12. Entering Table B under N = 12, number of P
events = 11 and number of P events = 12, and P = 0.50, we find
p(exactly 11 heads) = 0.0029
p(exactly 12 heads) = 0.0002
Therefore,
p(at least 11 heads) = p(11 or 12 heads) = 0.0029 + 0.0002 = 0.0031
c.
Because there are 12 coins, N = 12. Let P = the probability of getting a head in one flip of
any coin. Since the coins are unbiased, P = Q = 0.50. Since the problem asks for “as
extreme or more extreme than 11 heads,” the number of P events = 0, 1, 11, or 12 heads.
Entering Table B under N = 12, number of P events = 0, 1, 11, and 12, and P = 0.50, we find
p(exactly 0 heads)
p(exactly 1 head)
p(exactly 11 heads)
p(exactly 12 heads)
= 0.0002
= 0.0029
= 0.0029
= 0.0002
Therefore,
p(as extreme or more extreme than 11 heads)
d.
= p(0, 1, 11 or 12 heads)
= 0.0002 + 0.0029 + 0.0029 + 0.0002
= 0.0062
Because there are 12 coins, N = 12. Let P = the probability of getting a head in one flip of
any coin. Since the coins are biased such that the probability of a head with each is 0.40, P
= 0.40. The number of P events equals 12. Entering Table B under N = 12, number of P
events = 12 and P = 0.40, we find
p(exactly 12 heads) = 0.0000