A company is planning on expanding its reach. It is planning to open at least 22 more branches in the city and is considering 44 different locations for it. The probability of successful opening of a branch at a location is 0.650.65 and the success of branches are independent of each other.

Find the probability that exactly 22 branches will be successful.(Enter the answer upto 33 decimal accuracy.)

To solve this problem, we'll use the **binomial distribution**. The binomial distribution describes the probability of having exactly \( k \) successes in \( n \) independent Bernoulli trials, where each trial has a probability \( p \) of success.

### Problem Setup:
- Number of trials (locations considered): \( n = 44 \)
- Number of successes (successful branches): \( k = 22 \)
- Probability of success for each trial (branch being successful): \( p = 0.65 \)

### Binomial Probability Formula:

The probability of having exactly \( k \) successes in \( n \) trials is given by:

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

Where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \)
- \( p^k \) is the probability of \( k \) successes
- \( (1-p)^{n-k} \) is the probability of \( n-k \) failures

### Calculation:
We need to find \( P(X = 22) \):

\[
P(X = 22) = \binom{44}{22} \cdot (0.65)^{22} \cdot (1-0.65)^{44-22}
\]

Let's calculate this step by step.

#### 1. Calculate the binomial coefficient \( \binom{44}{22} \):

\[
\binom{44}{22} = \frac{44!}{22! \cdot 22!}
\]

#### 2. Calculate \( p^{22} \) and \( (1-p)^{22} \):

\[
p^{22} = (0.65)^{22}
\]
\[
(1-p)^{22} = (0.35)^{22}
\]

#### 3. Multiply all the components together.

I'll calculate this now.The probability that exactly 22 branches will be successful is approximately **0.015** (to three decimal places).

Would you like any further details on this calculation, or do you have any other questions?

Here are some related questions to consider:
1. What is the probability of having at least 22 successful branches?
2. How does the probability change if the success rate decreases to 0.55?
3. What is the expected number of successful branches?
4. How does the standard deviation of the number of successful branches change with different success probabilities?
5. What is the cumulative probability of having 22 or fewer successful branches?

**Tip:** Understanding the binomial distribution is crucial for scenarios involving a fixed number of independent trials with two possible outcomes.

Learn how to calculate the probability of exactly 22 successful branches using the binomial distribution. Understand the concepts of independent trials, success probability, and binomial coefficients in this advanced probability problem.

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