Question

You enter a karaoke contest. The singing order for the 22 contestants is randomly selected. what is the probability that you are not one of the first two singers?

Answer 1

To calculate the probability that you are not one of the first two singers, we need to determine the total number of possible outcomes and the number of favorable outcomes.

Total number of outcomes:

Since the singing order for the 22 contestants is randomly selected, the total number of possible outcomes is the number of ways to arrange all 22 contestants, which is given by 22!

Number of favorable outcomes:

To calculate the number of favorable outcomes, we consider that there are 20 remaining spots available after the first two singers have been chosen. The remaining 20 contestants can be arranged in 20! ways.

Therefore, the number of favorable outcomes is 20!

Now, let's calculate the probability:

Probability = Number of favorable outcomes / Total number of outcomes

Probability = 20! / 22!

To simplify this expression, we can cancel out common factors:

Probability = (20!)/(22×21×20!) = 1/ (22×21) = 1/462

Therefore, the probability that you are not one of the first two singers in the karaoke contest is 1/462.

Answer 2

82.25%

Explanation:

To calculate the probability that you are not one of the first two singers in a karaoke contest with 22 contestants, we need to determine the number of favorable outcomes and the total number of possible outcomes.

The number of favorable outcomes is the number of possible positions for you in the singing order after the first two positions are taken. Since the first two positions are fixed, there are 22 - 2 = 20 remaining positions available for you.

The total number of possible outcomes is the total number of ways to arrange all 22 contestants in the singing order, which is given by the factorial of 22 (denoted as 22!).

Therefore, the probability can be calculated as follows:

Probability = Number of favorable outcomes / Total number of possible outcomes

Number of favorable outcomes = 20! (arranging the remaining 20 positions for you)

Total number of possible outcomes = 22!

Probability = 20! / 22!

Now, let's calculate the probability using this formula:

Probability = (20 * 19 * 18 * ... * 3 * 2 * 1) / (22 * 21 * 20 * ... * 3 * 2 * 1)

Simplifying this expression, we find:

Probability = (20 * 19) / (22 * 21) = 380 / 462 ≈ 0.8225

Therefore, the probability that you are not one of the first two singers in the karaoke contest is approximately 0.8225 or 82.25%.