Imagine that you are applying for jobs.  The probability that a submitted application yeilds an interview is 20%.  If you apply for five jobs, what is the probability that you get less than two interviews?

To solve this problem, we can model it as a **binomial distribution**. In a binomial distribution, the probability of success (getting an interview) is \( p = 0.2 \), and the probability of failure (not getting an interview) is \( 1 - p = 0.8 \). You are applying for 5 jobs, so the number of trials is \( n = 5 \).

We need to find the probability of getting **less than two interviews**, which means getting either 0 or 1 interview. This is the sum of the probabilities for 0 interviews and 1 interview.

The probability mass function (PMF) of a binomial distribution is given by:

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

where:
- \( n \) is the number of trials,
- \( k \) is the number of successes (in this case, the number of interviews),
- \( p \) is the probability of success,
- \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes from \( n \) trials.

We need to calculate \( P(X = 0) \) and \( P(X = 1) \), then add them together.

### Step 1: Probability of 0 interviews \( (X = 0) \)
\[
P(X = 0) = \binom{5}{0} (0.2)^0 (0.8)^5 = 1 \cdot 1 \cdot (0.8)^5 = 0.32768
\]

### Step 2: Probability of 1 interview \( (X = 1) \)
\[
P(X = 1) = \binom{5}{1} (0.2)^1 (0.8)^4 = 5 \cdot 0.2 \cdot (0.8)^4 = 5 \cdot 0.2 \cdot 0.4096 = 0.4096
\]

### Step 3: Total probability
Now, we sum these probabilities to find the probability of getting less than two interviews:

\[
P(X < 2) = P(X = 0) + P(X = 1) = 0.32768 + 0.4096 = 0.73728
\]

Thus, the probability of getting less than two interviews is approximately **0.737 or 73.7%**.

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Would you like further details or have any questions?

Here are 5 follow-up questions you might find interesting:
1. What is the probability of getting exactly 2 interviews?
2. How would the probability change if you applied for 10 jobs instead of 5?
3. What is the expected number of interviews when applying for 5 jobs?
4. How would the calculation change if the probability of an interview were 30% instead of 20%?
5. Can we find the probability of getting at least one interview?

**Tip:** The binomial distribution is often used to model the number of successes in a fixed number of independent trials, such as flipping a coin or applying for jobs.

Imagine that you are applying for jobs. The probability that a submitted application yields an interview is 20%. If you apply for five jobs, what is the probability that you get less than two interviews?

Calculate the probability of receiving fewer than two interviews when applying for five jobs, with a 20% chance of success per application. Learn how to use the binomial distribution formula to determine the likelihood of 0 or 1 interviews, perfect for high school students.

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