Probability of guessing correctly atleast 7 out of 10 answers in a 'True' or 'False' test is equal to

Options

A. `(11)/(64)`
B. `(11)/(32)`
C. `(11)/(16)`
D. `(27)/(32)`

Explanation

To find the probability of guessing correctly at least 7 out of 10 answers in a 'True' or 'False' test, we can use the binomial distribution formula. Here’s the step-by-step solution:

### Step 1: Understand the problem
We need to find the probability of getting at least 7 correct answers out of 10. This means we need to calculate the probabilities for getting exactly 7, 8, 9, and 10 correct answers.

### Step 2: Define the parameters
In a binomial distribution:
- n = number of trials (in this case, n = 10)
- p = probability of success on each trial (for a 'True' or 'False' test, p = 1/2)
- q = probability of failure (q = 1 - p = 1/2)

### Step 3: Write the binomial probability formula
The probability of getting exactly r successes in n trials is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
Where \( \binom{n}{r} \) is the binomial coefficient.

### Step 4: Calculate the probabilities for r = 7, 8, 9, and 10
We need to calculate:
\[ P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) \]

1. **For r = 7:**
\[ P(X = 7) = \binom{10}{7} \left(\frac{1}{2}\right)^7 \left(\frac{1}{2}\right)^{3} = \binom{10}{7} \left(\frac{1}{2}\right)^{10} \]
\[ = 120 \cdot \frac{1}{1024} = \frac{120}{1024} \]

2. **For r = 8:**
\[ P(X = 8) = \binom{10}{8} \left(\frac{1}{2}\right)^8 \left(\frac{1}{2}\right)^{2} = \binom{10}{8} \left(\frac{1}{2}\right)^{10} \]
\[ = 45 \cdot \frac{1}{1024} = \frac{45}{1024} \]

3. **For r = 9:**
\[ P(X = 9) = \binom{10}{9} \left(\frac{1}{2}\right)^9 \left(\frac{1}{2}\right)^{1} = \binom{10}{9} \left(\frac{1}{2}\right)^{10} \]
\[ = 10 \cdot \frac{1}{1024} = \frac{10}{1024} \]

4. **For r = 10:**
\[ P(X = 10) = \binom{10}{10} \left(\frac{1}{2}\right)^{10} \left(\frac{1}{2}\right)^{0} = \binom{10}{10} \left(\frac{1}{2}\right)^{10} \]
\[ = 1 \cdot \frac{1}{1024} = \frac{1}{1024} \]

### Step 5: Sum the probabilities
Now, we add all these probabilities together:
\[ P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) \]
\[ = \frac{120}{1024} + \frac{45}{1024} + \frac{10}{1024} + \frac{1}{1024} \]
\[ = \frac{120 + 45 + 10 + 1}{1024} = \frac{176}{1024} \]
\[ = \frac{11}{64} \]

### Final Answer
The probability of guessing correctly at least 7 out of 10 answers in a 'True' or 'False' test is \( \frac{11}{64} \).

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