Final answer:
The probability that at least one of the five friends gets her own calculator when they pick them up in random order is approximately 63.33%.
Step-by-step explanation:
The problem you are referring to is concerned with the concept of probability. More specifically, it deals with the probability of a particular type of permutation known as a derangement, where no element appears in its original position. To calculate the probability that at least one of the five friends gets her own calculator, we can subtract the probability of no friend getting her own calculator (a complete derangement) from 1.
For five friends, the number of derangements (represented by the symbol !5) can be calculated using the formula
- !n = n! * (1/0! - 1/1! + 1/2! - 1/3! + ... + (-1)^n / n!)
For n = 5, the calculation yields:
- !5 = 5! * (1 - 1 + 1/2 - 1/6 + 1/24 - 1/120)
- !5 = 120 * (1 - 1 + 0.5 - 0.1667 + 0.0417 - 0.0083)
- !5 = 120 * (0.3667)
- !5 = 44 (rounded to the nearest whole number as derangements must be an integer)
Therefore, the probability of a complete derangement is 44/120, and thus the probability of at least one friend getting her own calculator is:
- P(at least one) = 1 - P(no one) = 1 - (44/120)
- P(at least one) = 76/120
- P(at least one) = 0.6333 (rounded to four decimal places)
The probability that at least one friend gets her own calculator is about 63.33%.