Sixty percent of a particular model of car are silver. What is the probability that in the next 10 observations of this model you observe 5 silver cars?

To calculate the probability of observing 5 silver cars out of the next 10 observations, we can use the binomial probability formula. The formula is:

P(x) = C(n, x) * p^x * q^(n-x)

Where:
- P(x) is the probability of observing exactly x successes,
- C(n, x) is the number of combinations of n items taken x at a time (also known as "n choose x"),
- p is the probability of success for a single observation, and
- q is the probability of failure for a single observation (q = 1 - p).

In this case, n = 10 (the number of observations), x = 5 (the number of successes, which is observing 5 silver cars), p = 0.6 (the probability of observing a silver car on a single observation), and q = 1 - p = 1 - 0.6 = 0.4.

Let's substitute these values into the formula:

P(5) = C(10, 5) * 0.6^5 * 0.4^(10-5)

C(10, 5) can be calculated as:
C(10, 5) = 10! / (5! * (10-5)!)
= 10! / (5! * 5!)
= (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1)
= 252

Substituting these values into the formula:

P(5) = 252 * 0.6^5 * 0.4^5

Calculating the final probability:

P(5) ≈ 0.2006581248, or approximately 0.2007 (rounded to four decimal places).

Therefore, the probability of observing 5 silver cars out of the next 10 observations is approximately 0.2007, or 20.07%.