) If a car agency sells 50% of its inventory of a certain foreign car equipped with side airbags. Find formula
for the probability distribution of the number of the cars with side airbags among the next 4 cars sold by
the agency and f(x)=1/16 (4/x), x=0,1,2,3,4

To find the formula for the probability distribution of the number of cars with side airbags among the next 4 cars sold by the agency, we need to recognize that this scenario can be modeled using the **binomial distribution**. Here's how to approach the problem:

### Given:
- The probability of a car being equipped with side airbags, \( p = 0.5 \) (since 50% of the cars have side airbags).
- The number of trials, \( n = 4 \) (since we're considering the next 4 cars sold).
- We need to find the probability distribution of \( X \), the number of cars with side airbags out of the 4 cars.

### The Binomial Distribution Formula:
The probability of getting exactly \( x \) successes (in this case, cars with side airbags) in \( n \) independent trials is given by the binomial distribution formula:

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

Where:
- \( \binom{n}{x} \) is the binomial coefficient, calculated as \( \binom{n}{x} = \frac{n!}{x!(n-x)!} \).
- \( p \) is the probability of success on each trial.
- \( x \) is the number of successes (cars with side airbags).

### Applying the Formula:
For \( n = 4 \) and \( p = 0.5 \):

\[
P(X = x) = \binom{4}{x} (0.5)^x (0.5)^{4-x} = \binom{4}{x} \left(\frac{1}{2}\right)^4 = \binom{4}{x} \frac{1}{16}
\]

Where \( x = 0, 1, 2, 3, 4 \).

### The Formula for \( f(x) \):
Given \( f(x) = \frac{1}{16} \binom{4}{x} \), the probability distribution can be written as:

\[
f(x) = \frac{1}{16} \times \frac{4!}{x!(4-x)!}, \quad x = 0, 1, 2, 3, 4
\]

Where \( \frac{4!}{x!(4-x)!} \) simplifies to:

- For \( x = 0 \): \( f(0) = \frac{1}{16} \times 1 \)
- For \( x = 1 \): \( f(1) = \frac{1}{16} \times 4 \)
- For \( x = 2 \): \( f(2) = \frac{1}{16} \times 6 \)
- For \( x = 3 \): \( f(3) = \frac{1}{16} \times 4 \)
- For \( x = 4 \): \( f(4) = \frac{1}{16} \times 1 \)

This results in the following probabilities:

\[
f(x) = \frac{1}{16} \times \{1, 4, 6, 4, 1\}, \quad x = 0, 1, 2, 3, 4
\]

Which gives:

\[
f(x) = \frac{1}{16} \{1, 4, 6, 4, 1\}
\]

### Conclusion:
The formula for the probability distribution \( f(x) \) for \( x \) cars with side airbags among the next 4 cars sold is:

\[
f(x) = \frac{1}{16} \times \binom{4}{x}, \quad x = 0, 1, 2, 3, 4
\]

Would you like further details or have any questions?

### Related Questions:
1. How does the binomial distribution change if the probability \( p \) of having side airbags increases to 0.7?
2. What is the expected number of cars with side airbags among the next 4 cars sold?
3. How would the probability distribution change if we were considering the next 5 cars instead of 4?
4. Can you explain the difference between a binomial and a Poisson distribution in this context?
5. How would you calculate the cumulative distribution function (CDF) for this binomial distribution?

### Tip:
Understanding binomial coefficients and their properties can greatly simplify calculating probabilities in binomial distributions.

Understand the probability distribution of cars with side airbags among the next 4 cars sold by an agency using the binomial distribution. Learn the binomial distribution formula and how to apply it to calculate probabilities. Suitable for students and professionals interested in probability theory and statistics.

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