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 first need to understand the concept of probability and then apply it to this specific scenario.
Probability is the likelihood or chance of an event occurring, and is expressed as a number between 0 and 1. In this case, the event is collecting all 5 different designs of glasses.
Let's break down the problem step by step:
Step 1: Determine the total number of possible outcomes
In each visit to the gas station, there are 5 possible designs of glasses that can be obtained. Therefore, the total number of possible outcomes per visit is 5.
Step 2: Calculate the number of ways to collect all 5 designs in 15 visits
To collect all 5 designs in 15 visits, you need to ensure that you obtain a different design in each visit. In the first visit, you have 5 choices. In the second visit, you have 4 remaining choices. In the third visit, 3 remaining choices, and so on. We can calculate this using the concept of permutations.
The formula to calculate the number of permutations is:
P(n, r) = n! / (n-r)!
where n represents the total number of items and r represents the number of items chosen.
In this case, n = 5 (the total number of designs) and r = 5 (the number of designs needed).
Using the formula, we can calculate the number of ways to collect all 5 designs in 15 visits:
P(5, 5) = 5! / (5-5)! = 5! / 0! = 5!
The factorial of a number is the product of all positive integers less than or equal to that number. Therefore, 5! = 5 x 4 x 3 x 2 x 1 = 120.
So, there are 120 ways to collect all 5 designs in 15 visits to the gas station.
Step 3: Calculate the total number of possible outcomes
To find the total number of possible outcomes in 15 visits, we multiply the number of possible outcomes per visit by the number of visits.
Total number of possible outcomes in 15 visits = (number of possible outcomes per visit) ^ (number of visits)
= 5^15
= 30,517,578,125
Step 4: Calculate the probability
Finally, we divide the number of ways to collect all 5 designs by the total number of possible outcomes to find the probability.
Probability = (number of ways to collect all 5 designs) / (total number of possible outcomes)
= 120 / 30,517,578,125
So, the probability of collecting all 5 different designs in 15 visits to the gas station is approximately 3.93 x 10^-9, or 0.00000000393, or about 1 in 254,313,177.