IanStewart wrote:
benjiboo wrote:
This is the binomial distribution concept.
2: We must reword as with most GMAT problems. We are really being asked about the probability of 0,1,2,3,4, or 5 heads (successes) all within the one 10trail run.
This is not a straightforward binomial distribution question. You cannot rephrase the question as you've done; we do not want to count all of the ways we might get five Heads, because sometimes when we get 5 Heads, two or more Heads will occur consecutively. We do not, for example, want to count the sequence HHHTTTHHTT.
If you complete the calculation as you suggest, you'd find the answer to be 319/512, or 62.3%, which is considerably too high.
And I'm not sure where the jpeg you posted comes from, but it contains a mistake; when the author refers to \(p\), he or she means to refer to \(\pi\).
I agree this is not a straight forward binomial distribution question. I think the formula needs to be modified for a question like this, which I tried to do. I am sure there is a mathematical answer to this using some form of a maniuplated binomial distribution equation. Lets discuss this together.
The question is asking you the number of places you can place a success given (0 heads given 10 tails) + (1 heads given 9 tails) + (2 heads 8 tails) + (3 heads 7 tails) + (4 heads 6 tails) + (5 heads 5 tails) in 10 trials of a binomial event, whereas a success is defined as a heads in any spot that is not next to another heads...
Meaning, for (1 head given 9 tails) you have _T_T_T_T_T_T_T_T_T_ 10 spots. You must take the 10 spots and choose 1 of them. How many different ways can you choose that 1 spot? There are 10c1 ways. Or 10!/(10-1)!*1!.]
As you see in the part you quoted of me, I said the "probability of getting 0,1,2,3,4, or 5 heads
(successes)" Therefore, the wording is correct, as I am eliminating the failures by not counting spots for them.
Also, if you do correctly use the equation I provided (my explanation tells you that in a case like this the first part of the equation is the SUM of the first part for each success), you get the right answer.
Here is the math:
First half of the equation:0 heads, 10 tails= TTTTTTTTTT = 0c0
1 Heads, 9 tails = _T_T_T_T_T_T_T_T_T_ = 10c1
2 heads, 8 tails = _T_T_T_T_T_T_T_T_ = 9c2
3 heads, 7 tails = _T_T_T_T_T_T_T_ = 8c3
4 heads, 6 tails = _T_T_T_T_T_T_ = 7c4
5 heads, 5 tails = _T_T_T_T_T_ = 6c5
0c0 + 10c1 + 9c2 + 8c3 + 7c4 + 6c5 =
1 + 10 + 36 + 56 + 35 + 6 =
144
second part of the equationThe second part of the formula, r becomes 144, and N is 10.
Why is r 144? Because as from above there are 144 chances of a success.(1/2)144 * (1/2)^(10-144) =
1/2^144 * 1/2^-134 =
1/2^144 * 2^134 =
2^-10 =
1/2^10 =
1/1024
Put it together
144 * 1/1024 =
144/1024 = answer
144/1024 = probability of getting 0,1,2,3,4, or 5 heads in a successful manor as defined by the question.
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I believe this is right. However I welcome all discussion.
My goal is to provide good and clear explanations for forum members. Sometimes somebody has already given a great explanation, so I try and find an alternative explanation to help further people's understandings.
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Benjiboo