Probability and Smarties
A little box contains 40 smarties: 16 yellow, 14 red and 10 orange.
You draw 3 smarties at random (without replacement) from the box.
What is the probabilty that you get 2 smarties of one color and another smartie of a different color?
Express your answer as a percentage rounded to the nearest whole percent.
The answer is 67.
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2 solutions
could you please explain to me how you got those answers in those brackets. for the life of me, I can't figure it out.
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( r n ) means n C r .
n C r = r ! ( n − r ) ! n !
See Combinatorics .
Toooo calculative!
(a) 2 yellow and 1 different color
P a = 4 0 C 3 ( 1 6 C 2 ) ( 2 4 C 1 ) = 2 4 7 7 2
(b) 2 red and 1 different color
P b = 4 0 C 3 ( 1 4 C 2 ) ( 2 6 C 1 ) = 3 8 0 9 1
(c) 2 orange and 1 different color
P c = 4 0 C 3 ( 1 0 C 2 ) ( 3 0 C 1 ) = 9 8 8 1 3 5
P = P a + P b + P c = 2 4 7 7 2 + 3 8 0 9 1 + 9 8 8 1 3 5 = 0 . 6 6 7 6 1 1 3 3 6
as a percentage rounded to the nearest whole percent,
P = ( 0 . 6 6 7 6 1 1 3 3 6 ) ( 1 0 0 ) = 6 7 %
You can write n C r as ^nC_r.
The number of ways to choose 2 yellow + 1 other smarties is ( 2 1 6 ) × 2 4 = 2 8 8 0
The number of ways to choose 2 red + 1 other smarties is ( 2 1 4 ) × 2 6 = 2 3 6 6
The number of ways to choose 2 orange + 1 other smarties is ( 2 1 0 ) × 3 0 = 1 3 5 0
The number of ways to choose 3 smarties is ( 3 4 0 ) = 9 8 8 0
The probability is 9 8 8 0 2 8 8 0 + 2 3 6 6 + 1 3 5 0 ≈ 0 . 6 6 7 6 1 1 3 3 6 .
Expressed as a percentage rounded to the nearest whole percent, this is 6 7 % .