Question 1200465: A coin is tossed 5 times. What is the probability that the number of heads obtained will be at most 2? Express your answer as a fraction or a decimal number rounded to four decimal places.
Answer by greenestamps(13214)   (Show Source):
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The total number of possible outcomes on the 5 flips is 2^5 = 32.
The number of ways of getting at most 2 heads is 5 choose 0 plus 5 choose 1 plus 5 choose 2: 
So the probability of getting at most 2 heads is 16/32 = 1/2.
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
1 5 10 10 5 1
So knowing the numbers in the 5th row of Pascal's Triangle makes solving this problem very easy.
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.
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.
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.
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