From a group of 12 people, you randomly select 3 of them. Find the probability that at least one of them is among the 3 oldest people in the group
To find the probability that at least one of the three randomly selected people is among the three oldest people in the group, we need to calculate the probability of the complementary event, which is the probability that none of the three selected people are among the three oldest.
Step 1: Calculate the total number of possible outcomes
Since we are selecting 3 people from a group of 12, the total number of possible outcomes is calculated using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of people in the group (12) and r is the number of people we are selecting (3).
So, the total number of possible outcomes is C(12, 3) = 12! / (3!(12-3)!) = 220.
Step 2: Calculate the number of outcomes where none of the three selected people are among the three oldest.
To calculate this, we need to consider that there are three oldest people in the group, and we need to select three people who are not among those three. So, we have 12 - 3 = 9 people to choose from.
Using the same combination formula, we calculate the number of possible outcomes where none of the three selected people are among the three oldest as C(9, 3) = 9! / (3!(9-3)!) = 84.
Step 3: Calculate the probability of the complementary event.
The probability of the complementary event (none of the three selected people are among the three oldest) is the number of outcomes where none of the three oldest are selected divided by the total number of possible outcomes.
So, the probability of the complementary event is 84/220.
Step 4: Calculate the probability of the desired event.
The probability that at least one of the three selected people is among the three oldest is the complement of the probability of the complementary event.
Therefore, the probability of at least one of the three selected people being among the three oldest is 1 - (84/220) = 136/220.
Simplifying, the probability is 34/55.
Hint: It's the complement of the probability that none of them is among that group.
Let En be the indexed event that the nth person selected is not one of the three oldest.
1 - P(E1) P(E2|E1) P(E3|E1 and E2)
= 1 - (9/12) (8/11) (7/10)
= 0.61818....