Math question

Question

Hi,

Can you please solve this?

"If the coin is tossed 5 times, what is the probability that at least 3 out of 5 times it will show heads?"

KarishmaB · Answer

If a coin is tossed 5 times, at each toss there are 2 possibilities H or T. So there are in all 2^5 = 32 possible outcomes e.g. HHHHH, HHTHH, HTTTH etc

Of these, we want those in which we get at least 3 heads.
So let us try and find the cases where we get 3/4/5 heads.

3 heads - we have to arrange HHHTT in as many different ways as possible e.g. HHTHT, TTHHH etc.
No of ways of arranging 5 letters 3 of which are same and 2 of which are same = 5!/3!2! = 10 ways

4 heads - Arrangements will be HHHHT, HHTHH etc
No of ways of arranging 5 letters 4 of which are same = 5!/4! = 5 ways

5 heads - there is only one such arrangement HHHHH

Hence probability of getting 3/4/5 heads = No of ways of getting 3/4/5 heads/Total no of ways = 16/32 = 1/2

Note: We don't really need to solve when we need 3/4/5 heads because we can   have 0/1/2/3/4/5 heads and correspondingly 5/4/3/2/1/0 tails. Since the cases are symmetrical, half of them will have 0/1/2 heads(and 5/4/3 tails) and the other half will have 3/4/5 heads (and 2/1/0 tails)

BTW, I think you need to post PS questions in the Quant - PS forum.

Duke007 · Answer

I am trying to help you but I am far to be a GMAT guru so it would be nice if someone more qualified could verify what I am saying.

How many ways are there to do 3 heads ou of 5? Is this the same to ask you how many words can be done with 3*h and 2*t

5!/2!3!= 5*4*3*2/2*3*2*

10* the number (of the probability to have any of the situation on the 5 shots=1/2^5) 2*2*2*2*2=32

10/32 or 1/16 is it possible?

pkonduri · Answer

Hi Karishma,

Are you not invalidating the choice of getting more that 3 heads , you are only considering hthth, why not hhhht and hhhhh that is 4 heads and 5 heads.

I feel answer should be 1/2

Duke007 · Answer

Suddenly your feeling makes me feel the same, I remember a section in the veritas prep books, to be careful with coins as the chance to have: H-h-t-t-t is actually the same as to have t-t-t-h-h...

Duke007 · Answer

However I believe an Veritas GMAT instructor knows what he says :-)

KarishmaB · Answer

Sorry, the GMAT instructor did not read the question properly... skipped 'at least'... that should be a lesson!!! 
I will edit the solution given above..

petrifiedbutstanding · Answer

Thanks for the solution.

But can you explain the post script note again? I don't understand it.

KarishmaB · Answer

Sure.
When you toss a coin 5 times, possibilities range from HHHHH, HHHHT, HHHTH...... TTTTT. 
The number of heads can be 0/1/2/3/4/5.

In these 32 arrangements, each arrangement will have either 3/4/5 Heads or 3/4/5 Tails (When there are 0/1/2 heads, the number of tails will be 5/4/3 respectively).
Since Heads and Tails are equal in status i.e. their probability of appearing is the same, whatever is the probability of 3/4/5 heads, same will be the probability of 3/4/5 tails. Together they account for probability 1 since every arrangement will have either 3/4/5 heads or 3/4/5 tails. So probability of each will be 1/2.

petrifiedbutstanding · Answer

Ah. I see. So essentially, what you're saying is, just by looking at the problem and doing some shrewd analysis, the solution is quite straightforward.

Thanks. This is a great approach.

144144 · Answer

Karishma- u have a skill of teaching... thanks. +1 again.

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