the coin tossed 10 times is it unusuall to get 2 heads 

To determine whether getting exactly 2 heads in 10 coin tosses is unusual, we need to calculate the probability using the binomial distribution and compare it to a significance threshold. Typically, an event is considered unusual if its probability is less than 5% (0.05).

Here, we have:
- \( n = 10 \) (number of trials)
- \( k = 2 \) (number of successes, or heads)
- \( p = 0.5 \) (probability of success in each trial)

The binomial probability formula is:
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \]

### Step-by-Step Calculation:

1. **Calculate the binomial coefficient** \(\binom{n}{k}\):
\[ \binom{10}{2} = \frac{10!}{2!(10 - 2)!} = \frac{10!}{2! \cdot 8!} = \frac{10 \cdot 9 \cdot 8!}{2 \cdot 1 \cdot 8!} = \frac{10 \cdot 9}{2 \cdot 1} = 45 \]

2. **Calculate the probability** \( p^k \):
\[ p^k = 0.5^2 = 0.25 \]

3. **Calculate the probability** \( (1 - p)^{n - k} \):
\[ (1 - p)^{n - k} = (1 - 0.5)^{10 - 2} = 0.5^8 = \frac{1}{256} \]

4. **Combine these values to find the probability**:
\[ P(X = 2) = \binom{10}{2} \cdot 0.5^2 \cdot 0.5^8 = 45 \cdot 0.25 \cdot \frac{1}{256} \]

\[ P(X = 2) = 45 \cdot \frac{0.25}{256} = 45 \cdot \frac{1}{1024} = \frac{45}{1024} \approx 0.043945 \]

### Conclusion:

The probability of getting exactly 2 heads in 10 coin tosses is approximately \( 0.043945 \) or \( 4.3945\% \).

Since \( 4.3945\% \) is less than the typical significance threshold of \( 5\% \), getting exactly 2 heads in 10 coin tosses can be considered unusual.

Would you like more details on this calculation or have any other questions?

Here are 8 related questions you might find interesting:

1. What is the probability of getting exactly 5 heads in 10 coin tosses?
2. How do you calculate the cumulative probability of getting at most 2 heads in 10 coin tosses?
3. What is the probability of getting more than 2 heads in 10 coin tosses?
4. How does the distribution of heads change as the number of tosses increases?
5. What is the expected number of heads in 10 coin tosses?
6. What is the variance and standard deviation of the number of heads in 10 coin tosses?
7. How would the probability change if the coin is biased with a probability of 0.6 for heads?
8. Can you explain the concept of the binomial distribution and its applications?

**Tip:** When dealing with probabilities in binomial distributions, it's useful to remember that small probabilities (less than 5%) often indicate unusual or rare events.

Calculate the probability of getting exactly 2 heads in 10 coin tosses using the binomial distribution formula. Understand why this event is considered unusual with a probability of approximately 4.3945%. Ideal for high school students studying probability and binomial distributions.

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