SOLUTION: There are 4 sets of balls numbered 1 through 5 placed in a bowl. If 4 balls are randomly chosen without replacement, find the probability that the balls have the same number. Expre

Question 1186814: There are 4 sets of balls numbered 1 through 5 placed in a bowl. If 4 balls are randomly chosen without replacement, find the probability that the balls have the same number. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

There are A = 5 ways to get what we want, since there are 5 items in the set {1,2,3,4,5}. This set is repeated four times to represent the 4*5 = 20 balls.

For example, one possible desired outcome is selecting the four balls labeled "3".
In other words, having this outcome: 3,3,3,3

There are n = 20 balls total and r = 4 of them are selected. Order doesn't matter, so we'll use the combination formula nCr

The notation is the same as C(n,r)

C(n, r) = (n!)/( r!*(n-r)! )
C(20, 4) = (20!)/( 4!*(20-4)! )
C(20, 4) = (20!)/( 4!*16! )
C(20, 4) = (20*19*18*17*16!)/( 4!*16! )
C(20, 4) = (20*19*18*17)/( 4! ) ..... the "16!" terms cancel
C(20, 4) = (20*19*18*17)/( 4*3*2*1 )
C(20, 4) = ( 116280 )/( 24 )
C(20, 4) = 4845
This represents the number of ways to select the four balls.
Let B = 4845

To summarize:
We have A = 5 ways to get what we want (having all four balls with the same number)
There are B = 4845 ways to select the four balls from a pool of 20

The probability of getting what we want is A/B = 5/4845 = 1/969. To reduce that fraction, I divided both parts by the GCF 5.

Use your calculator to find that 1/969 = 0.00103199174407 approximately. Rounding to the nearest millionth gets us 0.001032

Note: Rounding to the nearest millionths decimal place means the decimal answer will have 6 decimal places.

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Answer as a fraction: 1/969 (exact)
Answer as a decimal: 0.001032 (approximate; rounded to 6 decimal places)

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The solution from the other tutor shows finding the answer using the C(n,r) formula.

There is another basic method that obtains the answer with far less work.

Consider drawing the 4 balls from the bowl one at a time....

(1) first ball -- there are 20 balls in the bowl; we can pick any of them; P=20/20
(2) second ball -- there are 19 balls left, of which 3 match the first ball; P=3/19
(3) third ball -- there are 18 balls left, of which 2 match the first ball; P=2/18
(4) fourth ball -- there are 17 balls left, of which only 1 matches the first ball; P=1/17

The probability of drawing four matching balls is

ANSWER: 1/969

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