Question 313330: What is the probability of NOT tossing 3 heads with 3 fair coins?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! It would be 1 minus the probability of tossing 3 heads with 3 fair coins.
The probability of getting a head on one toss is equal to 1/2.
The probability of getting a tail on one toss is equal to 1/2.
The probability of tossing 3 heads with 3 fair coins is:
1/2 * 1/2 * 1/2
That probability becomes 1/8
the probability of NOT tossing 3 heads would then be 1 - 1/8 = 7/8
That probability can also be determined the hard way as follows:
If you don't toss 3 heads in a row, then you can toss:
0 heads out of 3 tosses, or 1 head out of 3 tosses, or 2 heads out of 3 tosses.
The probability of tossing 0 heads out of 3 tosses is the same as the probability of tossing 3 heads out of 3 tosses. That equals 1/8.
the probability of tossing 1 head out of 3 tosses is:
1/2 * 1/2 * 1/2 * 3 = 1/8 * 3 = 3/8
This itself is the probability of tossing:
htt or tht or tth
Since tossing 1 head out of 3 tosses can happen in 3 ways, and the probability for each of those ways is the same, you need to multiply the probability by 3.
The probability of htt is 1/2 * 1/2 * 1/2 = 1/8
The probability of tht is 1/2 * 1/2 * 1/2 = 1/8
the probability of tth is 1/2 * 1/2 * 1/2 = 1/8
The probability of tossing 2 heads out of 3 tosses becomes 3/8 as well.
You get:
hht = 1/2 * 1/2 * 1/2 = 1/8
hth = 1/2 * 1/2 * 1/2 = 1/8
thh = 1/2 * 1/2 * 1/2 = 1/8
Total is 3/8.
The probability of not getting 3 heads in 3 tosses is therefore.
p(0 heads) = 1/8
p(1 head) = 3/8
p(2 heads) = 3/8
Total probability of p(0 or 1 or 2) = p(0) + p(1) + p(2) = 1/8 + 2/8 + 3/8 = 7/8
The easy way was to take 1 minus the probability of 3 heads which we did up top.
You get the same answer.
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