Suppose an unfair coin comes up heads 52.2% of the time if it is flipped. If the coin is flipped 26 times, what is the probability that:
a) it comes up tails exactly 12 times?
b) it comes up heads more than 22 times?
This is a binomial expansion with p=0.522, and q=1-p=0.478.
We will calculate the terms of
(p+q)^26.
Using the notation
(n,r)=n!/((n-r)!r!)=n choose r
the binomial expansion can be expressed as
(p+q)^26
=p^26+(26,1)p^25q+(26,2)p^24q^2+...+(26,r)p^(26-r)q^r...+(26,1)pq^25+(26,0)q^26
P(12 tails)
=P(14 heads)
=(26,12)p^(12)q^(14)
=0.1285...
P(>22)
=P(23)+P(24)+P(25)+P(26)
=(26,3)p^23q^3+(26,2)p^24q^2+(26,1)p^25q+(26,0)p^26
=0.00001759...
To find the probabilities for these scenarios, we can use the binomial probability formula. The formula for the binomial probability is:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting k successes in n trials.
- C(n, k) is the number of combinations of n items taken k at a time.
- p is the probability of success in one trial.
- (1-p) is the probability of failure in one trial.
- n is the total number of trials.
- k is the number of successful outcomes we are interested in.
a) To find the probability that the coin comes up tails exactly 12 times out of 26 tosses, we can use the binomial probability formula with n=26, k=12, and p=0.478 (since the probability of tails is 1 - 0.522 = 0.478):
P(X=12) = C(26, 12) * 0.478^12 * (1-0.478)^(26-12)
To calculate the combination, C(26, 12), we use the formula:
C(n, k) = n! / (k! * (n-k)!)
Plugging in the values:
C(26, 12) = 26! / (12! * (26-12)!)
Now we can calculate each part of the equation and multiply them together to find the final probability.
b) To find the probability that the coin comes up heads more than 22 times out of 26 tosses, we need to calculate the sum of probabilities for k=23, 24, 25, and 26.
P(X > 22) = P(X=23) + P(X=24) + P(X=25) + P(X=26)
We use the same binomial probability formula for each individual probability calculation, with n=26, p=0.522, and k=23, 24, 25, 26. Then we sum up these probabilities to get the final answer.
I will now calculate these probabilities for you.