Equal Heads ?...

Probability Level 4

You and your brother are playing a game. Both of you have 6 fair coins each. Both of you throw all the coins simultaneously and count the number of heads each obtained with their 6 coins.

What is the probability that you obtain the exact same number of heads as your brother ?


The answer is 0.2255.

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Satyen Nabar by Satyen Nabar , 48, India

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1 solution

For each of the brothers the probability of getting k k heads in 6 6 throws is

( 6 k ) 2 6 \dfrac{\dbinom{6}{k}}{2^{6}} for 0 ≤ k ≤ 6 0 \le k \le 6 .

So, since the results of the brothers throws are independent of one another, the probability that both brothers get the same number of heads is

1 2 12 ∗ ∑ k = 0 6 ( 6 k ) 2 = 924 4096 = 231 1024 = 0.226 \dfrac{1}{2^{12}} * \displaystyle\sum_{k=0}^{6} \dbinom{6}{k}^{2} = \dfrac{924}{4096} = \dfrac{231}{1024} = \boxed{0.226}

to 3 3 decimal places.

15 Helpful 0 Interesting 0 Brilliant 0 Confused

Perhaps this problem could be even more interesting by replacing one of 12 12 fair coins by one biased coin. :)

Snehal Shekatkar - 6 years, 7 months ago

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