air coin is flipped 20 times.
a. Determine the probability that the coin
comes up tails exactly 15 times.
b. Find the probability that the coin comes
up tails at least 15 times.
c. Find the mean and standard deviation for
the random variable X giving the number
of tails in this coin flipping problem.
15
To solve these problems, we'll use the Binomial Distribution formula. The Binomial Distribution is applicable when there are two possible outcomes (in this case, heads or tails) and each outcome has a fixed probability of occurring (in this case, 1/2 since we assume the coin is fair).
a. To determine the probability that the coin comes up tails exactly 15 times, we'll use the formula:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
where:
- n is the number of trials (20 coin flips)
- k is the number of tails (15 tails)
- p is the probability of tails (1/2)
Plugging in the values into the formula:
P(X = 15) = (20 C 15) * (1/2)^15 * (1/2)^(20-15)
Using a combination formula (n C k) = n! / (k! * (n-k)!), we can calculate:
P(X = 15) = (20! / (15! * (20-15)!)) * (1/2)^20
Simplifying the expression:
P(X = 15) = 20! / (15! * 5!) * (1/2)^20
Evalutating the expression:
P(X = 15) = 0.0147857666 (approximately)
Therefore, the probability that the coin comes up tails exactly 15 times is approximately 0.0148.
b. To find the probability that the coin comes up tails at least 15 times, we need to calculate the sum of probabilities for all outcomes where the number of tails is greater than or equal to 15.
P(X >= 15) = P(X = 15) + P(X = 16) + ... + P(X = 20)
Using the formula from part (a), we can calculate each term and then sum them up to get the probability:
P(X >= 15) = Σ( (20 C k) * (1/2)^k * (1/2)^(20-k) ) [k = 15 to 20]
Evaluating the expression:
P(X >= 15) = 0.03515625 + 0.01953125 + 0.009765625 + 0.00439453125 + 0.001953125 + 0.000830078125
P(X >= 15) = 0.0716308594 (approximately)
Therefore, the probability that the coin comes up tails at least 15 times is approximately 0.0716.
c. The mean (or expected value) for a random variable X following a binomial distribution can be calculated using the formula:
μ = n * p
where:
- μ is the mean
- n is the number of trials (20 coin flips)
- p is the probability of tails (1/2)
Plugging in the values:
μ = 20 * (1/2)
μ = 10
Therefore, the mean for the random variable X is 10.
The standard deviation (σ) for a binomial distribution can be calculated using the formula:
σ = sqrt(n * p * (1-p))
Plugging in the values:
σ = sqrt(20 * (1/2) * (1 - 1/2))
σ = sqrt(20 * 1/2 * 1/2)
σ = sqrt(5)
Therefore, the standard deviation for the random variable X is sqrt(5).