Why can't I use binomial distribution for this

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dontknowmyuser
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A biased coin is tossed until a head appears or it is tossed four times. If P(Heads) = 2/3Write the probability distribution of the number of tosses.My question is why can't I use binomial distribution for each toss making x=1 and then x=2,3 and 4I worked it out another way and it worked I just want to know why binomial distribution can't be used for this question. Thanks
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mqb2766
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(Original post by dontknowmyuser)
A biased coin is tossed until a head appears or it is tossed four times. If P(Heads) = 2/3Write the probability distribution of the number of tosses.My question is why can't I use binomial distribution for each toss making x=1 and then x=2,3 and 4I worked it out another way and it worked I just want to know why binomial distribution can't be used for this question. Thanks
The "until a head appears" means the number of experiments, the "n" in the binomial distribution, is not constant for different outcomes. Or the probability tree is ragged - the branches have different lengths. You argue that you could continue the experiment to get a balanced tree, then aggregate the subtree under the first head. This would be correct, from the tree perspective (although inefficient). However, the binomial probability of 1 head appearing, is aggregated across different positions of the head in the full, balanced tree. Whereas, the probability of winning obviously depends on its position of the one head appearing. So the nCr part of the binomial is based on aggregating across the leaves of a balanced tree where order is unimportant. This is not the case here.
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dontknowmyuser
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(Original post by mqb2766)
The "until a head appears" means the number of experiments, the "n" in the binomial distribution, is not constant for different outcomes.
Or the probability tree is ragged - the branches have different lengths.
Thanks!!
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mqb2766
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(Original post by dontknowmyuser)
Thanks!!
Expanded the previous reply a bit.
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