The best chance is an illustration of probability
The 21st position in line gives you the best chance of being the first duplicate birthday
Assume you are in the nth position, the probability of getting a free ticket is:
P(n) = P'(first n - 1 people share a birthday) * P(birthday with first n -1 people)
Mathematically, the above can be represented as:
P(n) = [365/365 * 364/365 * 363/365 * ... * (365 - (n -2))/365] * [(n -1)/365]
Such that: n [tex]\le[/tex] 365
When simplified, the equation becomes
p(n)/p(n+1) = 365/(366 - n) * (n - 1)/n
Where
p(n)/p(n+1) > 1
So, we have:
365/(366 - n) * (n - 1)/n > 1
Evaluate the product
(365n -365)/(366n - n²) > 1
Cross multiply
365n -365 > 366n - n²
Collect like terms
n² + 365n - 366n - 365 > 0
Evaluate the difference
n² - n - 365 > 0
Solve for n using a graphing calculator;
n > - 18.6 or n >19.6
n cannot be negative.
So, we have:
n > 19.6
Approximate to nearest integer
n > 20
The least integer value of n is:
n = 21
Hence, the 21st position in line gives you the best chance of being the first duplicate birthday
Read more about probability at:
https://brainly.com/question/25870256
