Answer: The probability of getting exactly 16 heads when flipping 18 weighted coins, where the probability of a head on any one coin is 0.70, can be calculated using the binomial distribution.
The binomial distribution is used to calculate the probability of a specific number of successes in a fixed number of independent trials, where each trial has the same probability of success.
To calculate the probability of getting exactly 16 heads, we can use the formula:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting exactly k heads
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of success on a single trial (probability of getting a head)
- (1-p) is the probability of failure on a single trial (probability of getting a tail)
- n is the total number of trials (number of coins flipped)
- k is the number of successful trials (number of heads)
In this case, we have n = 18, p = 0.70, and k = 16.
Plugging in the values, we have:
P(X=16) = C(18, 16) * 0.70^16 * (1-0.70)^(18-16)
To calculate C(18, 16), we can use the formula:
C(n, k) = n! / (k! * (n-k)!)
Calculating C(18, 16):
C(18, 16) = 18! / (16! * (18-16)!)
Simplifying the expression:
C(18, 16) = 18! / (16! * 2!)
C(18, 16) = (18 * 17 * 16!) / (16! * 2)
C(18, 16) = (18 * 17) / 2
C(18, 16) = 153
Substituting the values back into the probability formula:
P(X=16) = 153 * 0.70^16 * (1-0.70)^(18-16)
Calculating P(X=16):
P(X=16) = 153 * 0.70^16 * 0.30^2
P(X=16) = 0.1771 (rounded to four decimal places)
Therefore, the probability of getting exactly 16 heads when flipping 18 weighted coins with a probability of a head of 0.70 is approximately 0.1771.