Answer:
the probability that your favorite song is one of the 20 that you hear that night is approximately 20 divided by the total number of possible outcomes.
Step-by-step explanation:
To find the probability that your favorite song is one of the 20 songs you hear that night, we need to consider the total number of possible outcomes and the number of favorable outcomes.
The total number of possible outcomes is the total number of ways to choose 20 songs out of the 800 available songs. This can be calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items chosen. In this case, n = 800 and r = 20.
Plugging in the values, we have:
C(800, 20) = 800! / (20!(800-20)!)
Simplifying this expression may be challenging due to the large numbers involved. However, we can use an approximation to calculate the probability.
Using Stirling's approximation for factorials:
n! ≈ √(2πn) * (n/e)^n
We can approximate the combination as follows:
C(800, 20) ≈ (√(2π(800)) * (800/e)^800) / [(√(2π(20)) * (20/e)^20) * (√(2π(780)) * (780/e)^780)]
The number of favorable outcomes is simply 20, as you want your favorite song to be one of the 20 songs you hear.
Therefore, the probability can be calculated as:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability ≈ 20 / C(800, 20)
While it is difficult to calculate the exact probability, we can use approximation techniques or numerical methods to find a close estimate.
Therefore, the probability that your favorite song is one of the 20 that you hear that night is approximately 20 divided by the total number of possible outcomes.
