8 coins are tossed simultaneously. The probability of getting at least 6 heads is

Options

A. `57/64`
B. `229/256`
C. `7/64`
D. `37/256`

Answer

To find the probability of getting at least 6 heads when 8 coins are tossed simultaneously, we can use the binomial probability formula. Let's break down the solution step by step.

### Step 1: Understand the problem
We need to find the probability of getting at least 6 heads when tossing 8 coins. This means we need to calculate the probabilities for getting exactly 6 heads, exactly 7 heads, and exactly 8 heads.

### Step 2: Define the parameters
In this case:
- Number of trials (n) = 8 (since we are tossing 8 coins)
- Probability of getting heads (p) = 1/2
- Probability of getting tails (q) = 1/2

### Step 3: Set up the binomial probability formula
The binomial probability formula is given by:
\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]
where \( \binom{n}{x} \) is the binomial coefficient, which represents the number of ways to choose x successes in n trials.

### Step 4: Calculate the probabilities
We need to calculate:
1. \( P(X = 6) \)
2. \( P(X = 7) \)
3. \( P(X = 8) \)

#### For \( P(X = 6) \):
\[ P(X = 6) = \binom{8}{6} \left(\frac{1}{2}\right)^6 \left(\frac{1}{2}\right)^{8-6} \]
\[ = \binom{8}{6} \left(\frac{1}{2}\right)^8 \]
\[ = 28 \times \frac{1}{256} = \frac{28}{256} \]

#### For \( P(X = 7) \):
\[ P(X = 7) = \binom{8}{7} \left(\frac{1}{2}\right)^7 \left(\frac{1}{2}\right)^{8-7} \]
\[ = \binom{8}{7} \left(\frac{1}{2}\right)^8 \]
\[ = 8 \times \frac{1}{256} = \frac{8}{256} \]

#### For \( P(X = 8) \):
\[ P(X = 8) = \binom{8}{8} \left(\frac{1}{2}\right)^8 \left(\frac{1}{2}\right)^{8-8} \]
\[ = \binom{8}{8} \left(\frac{1}{2}\right)^8 \]
\[ = 1 \times \frac{1}{256} = \frac{1}{256} \]

### Step 5: Sum the probabilities
Now, we can find the total probability of getting at least 6 heads:
\[ P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) \]
\[ = \frac{28}{256} + \frac{8}{256} + \frac{1}{256} \]
\[ = \frac{28 + 8 + 1}{256} = \frac{37}{256} \]

### Final Answer
The probability of getting at least 6 heads when tossing 8 coins is:
\[ \frac{37}{256} \]