company is planning on expanding its reach. It is planning to open at least 2 more branches in the city and is considering 4 different locations for it. The probability of successful opening of a branch at a location is 0.65 and the success of branches are independent of each other.

Find the probability that exactly 2 branches will be successful.(Enter the answer upto 3 decimal accuracy.)

We will use the **binomial distribution** again to find the probability that exactly 2 branches out of 4 will be successful.

### Problem Setup:
- Number of trials (locations considered): \( n = 4 \)
- Number of successes (successful branches): \( k = 2 \)
- Probability of success for each trial (branch being successful): \( p = 0.65 \)

### Binomial Probability Formula:

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

For this problem:

\[
P(X = 2) = \binom{4}{2} \cdot (0.65)^2 \cdot (1-0.65)^{4-2}
\]

I'll calculate this now.The probability that exactly 2 branches will be successful out of 4 is approximately **0.311** (to three decimal places).

Do you have any more questions, or would you like further details on this calculation?

Here are some related questions you might find interesting:
1. What is the probability that at least 2 branches will be successful?
2. What is the probability that all 4 branches will be successful?
3. What is the probability of having fewer than 2 successful branches?
4. How does the probability change if the success rate drops to 0.5?
5. What is the expected number of successful branches out of 4?

**Tip:** When dealing with small sample sizes, the binomial distribution can be easily computed by hand, which helps in understanding the underlying concepts better.

Learn how to calculate the probability that exactly 2 out of 4 branches will be successful using the binomial distribution. This problem involves applying the binomial probability formula with a success probability of 0.65. Approximate answer provided is 0.311 to three decimal places.

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