Thirty students are asked to pick a number between 1 and 60. Now find the probability that at least 2 will choose the same number

Answer 1

Each student picks a number between 1 and 60, so there are 60 possible choices for each student. Since there are 30 students, the total number of possible outcomes (without any restrictions) is:

60³⁰

Instead of directly calculating the probability that at least two students pick the same number (which is complex), we will first find the probability that all students pick different numbers, and then subtract this from 1 to find the desired probability.

Case where all students choose different numbers

1. For the first student, there are 60 possible choices.
2. For the second student, there are 59 remaining choices (since the second student cannot choose the same number as the first student).
3. For the third student, there are 58 remaining choices, and so on.

Thus, the number of favorable outcomes (where all 30 students pick different numbers) is:

60×59×58×⋯×(60−29)=60!/30!

The probability that all 30 students pick different numbers is the ratio of favorable outcomes to the total number of possible outcomes:

P(all different)=60×59×58×⋯×31/(60³⁰)

The probability that at least two students pick the same number is the complement of the probability that all students pick different numbers:

P(at least two the same)=1−P(all different)

Now we can express this as:

P(at least two the same)=1−(60×59×58×⋯×31)/(60³⁰)

This is the exact expression for the probability that at least two students choose the same number. You can approximate it.