To determine the probability of obtaining tails up at least two times when a fair coin is flipped seven times, we'll approach the problem step-by-step using properties of binomial distributions.
### Step 1: Define the Parameters
- n: Number of trials (coin flips) = 7
- p: Probability of success (getting tails in a single flip) = 0.5
### Step 2: Find the Complement
Instead of directly calculating the probability of getting at least two tails, we will find the complement, which is the probability of getting fewer than two tails (i.e., 0 or 1 tail), and then subtract that from 1.
### Step 3: Calculate the Probability of Getting 0 Tails
The probability of getting 0 tails in 7 trials can be determined by:
[tex]\[
P(X = 0) = \binom{7}{0} (0.5)^0 (0.5)^7 = 1 \cdot 1 \cdot (0.5)^7 = 0.5^7 = \frac{1}{128} \approx 0.0078125
\][/tex]
### Step 4: Calculate the Probability of Getting 1 Tail
The probability of getting exactly 1 tail in 7 trials can be determined by:
[tex]\[
P(X = 1) = \binom{7}{1} (0.5)^1 (0.5)^6 = 7 \cdot 0.5 \cdot (0.5)^6 = 7 \cdot 0.5 \cdot \frac{1}{64} = \frac{7}{128} \approx 0.0546875
\][/tex]
### Step 5: Add the Probabilities of Getting 0 or 1 Tail
[tex]\[
P(X < 2) = P(X = 0) + P(X = 1)
\][/tex]
[tex]\[
P(X < 2) = 0.0078125 + 0.0546875 = 0.0625
\][/tex]
### Step 6: Find the Probability of Getting At Least 2 Tails
The probability of getting at least 2 tails is the complement of getting fewer than 2 tails:
[tex]\[
P(X \geq 2) = 1 - P(X < 2)
\][/tex]
[tex]\[
P(X \geq 2) = 1 - 0.0625 = 0.9375
\][/tex]
### Step 7: Convert the Probability to Fraction
Since 0.9375 is equivalent to the fraction representation:
[tex]\[
0.9375 = \frac{15}{16}
\][/tex]
Therefore, the answer is:
B. [tex]\(\boxed{\frac{15}{16}}\)[/tex]
