Question 739596: From a group of 11 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.
I tried PA=1-all three persons selected are not the 3 oldest.
1-(10/11)(9/11)(8/11),but it is incorrect.
I found a similar problem,http://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.468918.html but I didn't get it after reading the solution.
Thanks for help!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! There are 11 C 3 = 165 ways to choose a group of 3 people (from a pool of 11)
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Note: I'm using the n C r function found on the calculator. You can find a calculator for that here or you can use the nCr function found on all TI-83/84/86/89/93 calculators (and found elsewhere too). More information about the nCr function and what it means is found here
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Now say that persons A, B and C are the three oldest (in that order). The other 8 persons are persons D through K. We can imply anything we want about persons D through K, but we can definitively say that persons D through K are all younger than A, B and C.
There are 8 C 3 = 56 ways to choose a group where NONE of the three oldest (A, B or C) show up. So there are 56 ways to pick a 3 person group from this pool: {D, E, F, G, H, I, J, K}
So there are 165 - 56 = 109 groups where persons A, B or C show up. This is either where A, B or C show up individually (2 members of D through K show up). Or this is where two of the oldest show up in the three person group (1 person who is D through K shows up). Or all 3 of the oldest show up: A, B, C.
So again, there are 109 groups where at least one member is A, B or C.
This is out of 165 groups total.
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So the probability is...
Fraction Form: 109/165
Decimal Form: 0.66060606060607 (this is approximate)
Percentage Form: 66.060606060607 % (this is approximate)
*** Note 1: to get the fraction form answer, you divide the number of groups that have the qualities you desire (109) over the number of groups total (165) ***
*** Note 2: To find the decimal form, you use a calculator to divide the fraction in the fraction form. To get the percentage form, you multiply the decimal form by 100 and you stick on a % sign. ***
*** Note 3: If you decide to go with either the decimal or percentage form, then don't forget to follow all instructions when it comes to rounding (eg: they might want you to round to the nearest hundredth, so pay close attention to that) ***
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