Tricky probability question

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In summary, the probability of obtaining exactly 1 head is 3/7, the probability of obtaining exactly 2 heads is 3/8, and the probability of obtaining exactly 3 heads is 1/8. There is a flaw in the first method as it only counts the outcomes with exactly 1 head, and in the second method, the probability is conditioned on obtaining at least one head. The unconditional probability of obtaining exactly 2 heads is 3/8.
  • #1
fraggle
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A coin is tossed 3 times. at least 1 head is obtained. Determine each probability...?

1) exactly 1 head is obtained
2) exactly 2 heads are obtained
3) exactly 3 heads are obtained.
------------------------------------------------------------

for 1) brute force indicates that there 7 possible combinations:

HHH, HHT, HTH, HTT, THH, THT, TTH (because at least one heads is obtained).

Out of these we see that there are 3 occasions where there is exactly one heads.
If we let S be the sample space of all possable outcomes, and let E be the event of having exactly one heads, then a basic rule of probability indicates that the probability P(E)=|E|/|S|=3/7.

However, using another method:
fix one coin throw's result as heads. Then the probability of the other two throws being tails is (1/2)(1/2)=1/4.
Since we can fix heads 3 times, this indicates that the probability is 3(1/4)=3/4.

Can anyone spot where the flaw is in either of these attempts at a solution?
thanks
(this isn't a homework question btw)
 
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  • #2
3/7 is a conditional probability given at least one head. For the unconditional probability you need to count TTT among the outcomes.

3/4 is the conditional probability of two tails given (one) head. "Head" has probability 1/2, so the unconditional probability is 3/4 x 1/2 = 3/8.
 

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