the roller coaster has two seats in each of 12 rows. Riders are assigned to seats in the order they arrive. if you ride this roller coaster once, what is the probablilty of getting coveted first row? How many times must you ride in order to have atleast a 95% chance of getting a first row seat atleast once?
If all seats fill every time, then the probability of getting a front row seat is 1/12.
To have a 95% chance if getting a front row seat, you must reduce the probabilty of not getting a seat to 0.05
(11/12)^N = 0.05
N log (11/12) = log .05
solve for N
Solve for N
To find the probability of getting a coveted first row seat on the roller coaster, we need to consider the total number of possible seats and the number of first-row seats.
The roller coaster has two seats in each of the 12 rows, so the total number of seats is 2 seats/row × 12 rows = 24 seats. Out of these, only the first row is coveted, which means there is only 1 coveted seat.
Therefore, the probability of getting a coveted first-row seat on a single ride of the roller coaster is:
Probability = Number of coveted seats / Total number of seats
= 1 seat / 24 seats
= 1/24
= 0.0417 (rounded to four decimal places)
To find out the number of times you must ride the roller coaster in order to have at least a 95% chance of getting a first-row seat at least once, we can use the concept of complementary probability.
The complementary probability is the probability of the event not occurring. In this case, it is the probability of not getting a first-row seat after a certain number of rides.
Let's assume you ride the roller coaster n times. The probability of not getting a first-row seat on a single ride is 23/24 (since there is only 1 coveted seat out of 24 seats).
Therefore, the probability of not getting a first-row seat after n rides is:
Probability of not getting a first-row seat on a single ride = (23/24)^n
To find the number of times you must ride for a 95% chance, we need to solve the following inequality:
(23/24)^n ≤ 0.05
To solve this inequality, you can use logarithms or trial and error. Taking the logarithm base 10 of both sides, we get:
log((23/24)^n) ≤ log(0.05)
n × log(23/24) ≤ log(0.05)
n ≥ log(0.05) / log(23/24)
Using a calculator, we find that log(0.05) / log(23/24) is approximately 183.19.
Therefore, you must ride the roller coaster at least 184 times to have at least a 95% chance of getting a first-row seat at least once.