Probability of Coin Tosses and Pascal’s Triangle

A coin’s faces are marked with 0 and 1. What is the probability of scoring K 1s in N tosses?

Let’s start with 3 tosses (N=3) and we want to find out the probability of scoring 2 (K=2) 1s.  Let’s map out the possible outcomes of three coin tosses.

N Possible Outcomes Possibility of
scoring K 1s
2 00 01 10 11 0: 1
1: 2
2: 1
3 000 001 010 011 100 101 110 111 0: 1
1: 3
2: 3
3: 1
4 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 0: 1
1: 4
2: 6
3: 4
4: 1

And if you noticed, that exactly looks like Pascal’s triangle! Just get the Kth entry from the Pascal’s triangle’s Nth row.

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