An honest coin is tossed 11 times in a row. Find the probability of Exactly one of the 11 tosses comes up​ tails

To find the probability that exactly one of the 11 tosses of an honest coin comes up tails, we can use the binomial probability formula.

The binomial probability formula is:

\[
P(k \text{ successes}) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \( n = 11 \) (the number of tosses),
- \( k = 1 \) (the number of tails, which is a "success" in this case),
- \( p = \frac{1}{2} \) (the probability of getting tails on each toss, since the coin is fair).

### Step 1: Calculate the binomial coefficient \(\binom{n}{k}\)

The binomial coefficient \(\binom{11}{1}\) represents the number of ways to choose 1 tail out of 11 tosses. This is given by:

\[
\binom{11}{1} = \frac{11!}{1!(11-1)!} = 11
\]

### Step 2: Calculate the probability for the event

The probability of exactly 1 tail is:

\[
P(1 \text{ tail}) = \binom{11}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^{11-1}
\]

\[
P(1 \text{ tail}) = 11 \times \left(\frac{1}{2}\right)^{11}
\]

\[
P(1 \text{ tail}) = 11 \times \frac{1}{2048} = \frac{11}{2048}
\]

### Final Answer:

\[
P(1 \text{ tail}) = \frac{11}{2048} \approx 0.00537
\]

So, the probability of getting exactly one tail in 11 tosses is \(\frac{11}{2048}\) or approximately 0.00537.

Would you like more details, or have any other questions?

Here are 5 related questions for further exploration:
1. What is the probability of getting exactly two tails in 11 tosses?
2. What is the probability of getting no tails in 11 tosses?
3. What is the probability of getting at least one tail in 11 tosses?
4. How does the probability change if the number of tosses increases to 12?
5. What is the expected number of tails in 11 tosses?

**Tip:** The sum of probabilities for all possible outcomes in a binomial experiment always equals 1.

An honest coin is tossed 11 times in a row. Find the probability of exactly one of the 11 tosses comes up tails.

Calculate the probability of getting exactly one tail in 11 coin tosses using the binomial probability formula. Explore key probability concepts and learn the steps to determine the exact probability, suitable for students in Grades 9-12.

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