A gas station is giving away free glasses with 5 different designs. What is the probability of collecting all 5 in 15 visits to the gas station?
To find the probability of collecting all 5 different designs in 15 visits to the gas station, we can use the concept of probability and combinatorics.
First, let's find the number of ways to collect all 5 designs in 15 visits. We can represent this as permutating the 5 designs over 15 visits. The number of ways to do this can be calculated using the formula for permutations: nPr = n! / (n - r)!, where n is the total number of items and r is the number of items selected in each permutation.
In this case, n is 15 (number of visits) and r is 5 (number of designs to collect). So, the number of ways to collect all 5 designs in 15 visits is:
15P5 = 15! / (15 - 5)! = 15! / 10! = (15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1) = 3003
Next, let's find the total number of outcomes or possibilities. In each visit, there are 5 different designs that can be received. So, the total number of outcomes for each visit is 5.
Therefore, the total number of outcomes or possibilities for 15 visits is:
5^15 = 30517578125
Finally, we can calculate the probability of collecting all 5 designs in 15 visits by dividing the number of ways to collect all 5 designs by the total number of outcomes:
Probability = (Number of ways to collect all 5 designs) / (Total number of outcomes) = 3003 / 30517578125 = 0.00000983 (approximately)
Hence, the probability of collecting all 5 designs in 15 visits to the gas station is approximately 0.00000983 or 0.000983%.