Water Balloon Fight

Probability Level 2

7 people are having a water balloon fight. At the same time, each of the 7 people throws a water balloon at one of the other 6 people, chosen at random. What is the probability that there are 2 people who throws balloons at each other?

17.3 23.5 55.7 50.5

Shiv Kumar by Shiv Kumar , 32, India

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1 solution

Hongqi Wang
Dec 8, 2020

  • each person has 6 choices, so total choices is S = 6 7 \\ S = 6^7

  • pick up two persons as a pair, other 5 can choose freely: S 1 = ( 2 7 ) ⋅ 6 5 \\ S_1 = \tbinom{2}{7} \cdot 6^5

  • pick 4 persons as 2 pairs, S 2 = ( 4 7 ) ⋅ 3 ⋅ 6 3 \\ S_2 = \tbinom{4}{7} \cdot 3 \cdot 6^3

  • pick up 6 for 3 pairs, S 3 = ( 6 7 ) ⋅ 5 ⋅ 3 ⋅ 6 1 \\ S_3 = \tbinom{6}{7} \cdot 5 \cdot 3 \cdot 6^1

So the possibility with pair is: P = S 1 − S 2 + S 3 S = ( 2 7 ) ⋅ 6 5 − ( 4 7 ) ⋅ 3 ⋅ 6 3 + ( 6 7 ) ⋅ 5 ⋅ 3 ⋅ 6 1 6 7 ≈ 0.5046 \\ P = \frac {S_1 - S_2 + S_3}S \\ = \frac {\tbinom{2}{7} \cdot 6^5 - \tbinom{4}{7} \cdot 3 \cdot 6^3 + \tbinom{6}{7} \cdot 5 \cdot 3 \cdot 6^1}{6^7} \\ \approx 0.5046

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