Question 1200465
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The total number of possible outcomes on the 5 flips is 2^5 = 32.<br>
The number of ways of getting at most 2 heads is 5 choose 0 plus 5 choose 1 plus 5 choose 2: {{{C(5,0)+C(5,1)+C(5,2)=1+5+10=16}}}<br>
So the probability of getting at most 2 heads is 16/32 = 1/2.<br>
For this kind of problem, involving flipping a fair coin n times, a familiarity with Pascal's Triangle is useful.  The entries in the 5th row of Pascal's Triangle are C(5,0), C(5,1), ..., C(5,4), and C(5,5).  Those numbers are<br>
1 5 10 10 5 1<br>
So knowing the numbers in the 5th row of Pascal's Triangle makes solving this problem very easy.<br>
A further understanding of this general topic allows you to find the answer of 1/2 without doing any calculations, and without using Pascal's Triangle.<br>
By symmetry, the probability of getting 3 heads is the same as the probability of getting 2 heads; the probability of getting 4 heads is the same as the probability of getting 1 head; and the probability of getting 5 heads is the same as the probability of getting 0 heads.<br>
So the probability of getting 0, 1, or 2 heads is then the same as the probability of getting 3, 4, or 5 heads; therefore the probability of getting at most 2 heads is 1/2.<br>