If ten coins are tossed together, then the probability of getting at least 7 heads is equal to:

Question

Accepted Answer

To determine the probability of getting at least 7 heads when ten coins are tossed, we need to calculate the probability of getting exactly 7, 8, 9, and 10 heads and then sum these probabilities.

The probability of getting exactly $ k $ heads in $ n $ coin tosses is given by the binomial probability formula:

$$
P(X = k) = \binom{n}{k} \left(\frac{1}{2}\right)^n
$$

where $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$.

For this problem, $ n = 10 $.

1. **Probability of exactly 7 heads:**
   $$
   P(X = 7) = \binom{10}{7} \left(\frac{1}{2}\right)^{10} = \frac{120}{1024}
   $$

2. **Probability of exactly 8 heads:**
   $$
   P(X = 8) = \binom{10}{8} \left(\frac{1}{2}\right)^{10} = \frac{45}{1024}
   $$

3. **Probability of exactly 9 heads:**
   $$
   P(X = 9) = \binom{10}{9} \left(\frac{1}{2}\right)^{10} = \frac{10}{1024}
   $$

4. **Probability of exactly 10 heads:**
   $$
   P(X = 10) = \binom{10}{10} \left(\frac{1}{2}\right)^{10} = \frac{1}{1024}
   $$

Now, we sum these probabilities to find the probability of getting at least 7 heads:

$$
P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)
$$
$$
= \frac{120}{1024} + \frac{45}{1024} + \frac{10}{1024} + \frac{1}{1024}
$$
$$
= \frac{176}{1024}
$$

Simplifying $\frac{176}{1024}$ gives us $\frac{11}{64}$.

Thus, the correct answer is Option 4: $\frac{11}{64}$.

**Explanation for Incorrect Options:**

- **Option 1: 175/1024** - This option is incorrect because the sum of probabilities for at least 7 heads is 176/1024, not 175/1024.

- **Option 2: 76/1024** - This option is incorrect because it significantly underestimates the probability. The correct sum of probabilities is 176/1024.

- **Option 3: 166/1024** - This option is incorrect because it underestimates the probability. The correct sum is 176/1024.

Therefore, based on the calculations, Option 4 is indeed the correct answer, as it accurately represents the probability of getting at least 7 heads when tossing ten coins.