If a child flips a coin five times in a row, what is the probability

Question

If a child flips a coin five times in a row, what is the probability that she will receive at least one head and one tail?

A) 3/4
 B) 11/12
 C) 15/16
 D) 31/32
 E) 63/64

Guys please help me. i know only one way which is a really long method wherein you list all possibilities but given the time constraints, its not possible. please show a shorter and easy way to solve this please

KarishmaB · Accepted Answer

The coin is flipped 5 times in a row. 
So there are 2*2*2*2*2 = 32 different outcomes possible such as HHTHH, HTHHT, TTHTH etc

Look for the reverse probability - when will there is no heads or no tails?
There will be no heads when you get all tails. There is only 1 such case TTTTT
There will be no tails when you get all heads. There is again only 1 such case HHHHH
So, out of 32, there are only 2 cases in which there are either no heads or no tails. In all other 30 cases, there is at least one heads and at least one tails.

Required probability = 30/32 = 15/16

Answer (C)

chetan2u · Answer

Hi,
you don't require to write down all possiblities but do the reverse as also suggested by karishma..
there are two ways where we do not have atleast one each of heads and tail and these are HHHHH and TTTTT..
prob of HHHHH= 1/2*1/2*1/2*1/2*1/2=\frac{1}{32} ...
similarly for TTTTT= 1/32..
therefore the prob of not getting one of each =1/32+1/32=1/16..
therefore prob of getting atleast one of each=1-1/16=15/16..

nikdiablo129 · Answer

Thank you so much maam. I give my gmat 2nd attempt on 22nd i have been practicing the 6 tests from Veritas prep

KarishmaB · Answer

Wishing you the best!

EMPOWERgmatRichC · Answer

Hi nikdiablo129,

When dealing with probabilities, there are two results that can be calculated: what you WANT to have happen and what you DON'T WANT to have happen. In certain questions, it's actually faster to calculate what you DON'T WANT, since....

(Probability of what you WANT) + (Probability of what you DON'T WANT) = 1

Keep an eye out for the second option. Calculating what you DON'T WANT and then subtracting that fraction from 1 might be the faster/easier option.

GMAT assassins aren't born, they're made,
Rich

sun01 · Answer

Teacher,

I have one query. The question writes to find the probability of at least 0ne head and one tail. I understood the process explained above. But the confusion arises from the word 'And' between two events. As per concept, “AND” means MULTIPLICATION of two independent events.

Please clear my doubts.

Regards,

Sun01

KarishmaB · Answer

AND is one of the most commonly used words and it is not possible that it should be used only when you need to multiply the probability of two independent events. The concept is that if there are two independent events and you need to find the probability that one happens "and" the other happens, then you multiply the probabilities. Note that the actual question may not even use "and". It may use "as well as" or "additionally" etc. Focus on the logic of the question.

sowragu · Answer

In probability, whenever you see "ATLEAST" or "ATMOST" in the question stem choose for the reverse option. That will make things simpler.

If question is saying at least one head in 3 flips means the reverse is zero heads which is all tails.

Hope this clears.

iliavko · Answer

VeritasPrepKarishma

What if we convert the H-T event into a single event?

Then we would get 4 possible rolls and we could subtract 1-P(of HT not happening) that would be 16/16-1/16?

Like this makes sense, but for me there is a problem to understand what exactly "HT not happening" means in the new context of 4 rolls and HT being a single event.

Can this technique be used here? Or its too confusing?

Thank you!

KarishmaB · Answer

Yes, the logic is that if you have at least one heads and at least one tails, there will be at least one pair of HT or TH when you write down the 5 rolls.

e.g. HHH HT 
 TH HTH
etc

Every pair is either HH, TT, HT or TH so probability of HT or TH is 1/2.

Starting from the first roll, in HHHHT, you get 4 pairs: HH, HH, HH, HT

Probability that all for pairs are HH or TT is 1/2 * 1/2 * 1/2 * 1/2 = 1/16

Probability that you have at least one HT or TH pair = 1 - 1/16 = 15/16

But this method just complicates a simple question.

iliavko · Answer

Thank you so much, Karishima! As you said, indeed the conversion to a single event only complicated this problem! Thank you for your input

anuj11 · Answer

i tried solving the problem by using the formula : - Total = All heads + (At least 1 tail), the following equation resulted in 31/32

Probability of all heads = 1, so at least 1 tail would be 1- 1/32 = 31/32. (at least 1 tail)

Similarly for heads

Though i did guess the right answer i would like to know whats wrong with my approach

EMPOWERgmatRichC · Answer

Hi anuj11,

I think that there might be some 'typos' in what you wrote.

When flipping a coin 5 times, there are only 2 outcomes that do not include at least one head and at least one tail....

To start, the probability of getting "all heads" is (1/2)(1/2)(1/2)(1/2)(1/2) = 1/32 and the same probability exists for "all tails" (re: 1/32).

Thus, the probability of "all heads" or "all tails" = (1/32) + (1/32) = 2/32 = 1/16

Taking all of this one step further, the probability of getting "AT LEAST one head AND AT LEAST one tail" = 1 - 1/16 = 15/16

GMAT assassins aren't born, they're made,
Rich

BrentGMATPrepNow · Answer

If a coin is flipped 5 times in a row, there are  32  possible outcomes (HHTTH, HTHTT, TTTTT, etc) 
How did we get  32 ?
Well, there are 2 possible outcomes for the 1st coin flip, 2 possible outcomes for the 2nd coin flip, 2 outcomes for the 3rd flip, 2 outcomes for the 4th flip, and 2 outcomes for the 5th flip, 
By the Fundamental Counting Principle (FCP), the TOTAL number of outcomes when flipping 5 coins = (2)(2)(2)(2)(2) =  32

Of course, among those  32  outcomes, there are some outcomes that DO NOT meeting the required condition that we receive  at least one head and one tail 
 There are  EXACTLY  2  outcomes that DO NOT meet this required condition.  
They are: TTTTT and HHHHH

So, the number of outcomes that DO meet the required condition =  32  -   2   =   30

So, P(child gets at least one head and one tail) =   30  / 32 = 15/16

Answer: C

Cheers,
Brent

Bismarck · Answer

OA: C

The probability that she will receive at least one head and one tail,If a child flips a coin five times in a row = 1 - Probability of all 5 heads in row - Probability of all 5 tails in row 
 = 1- (\frac{1}{2})^5- (\frac{1}{2})^5 
 = 1- 2*(\frac{1}{2})^5 = 1-(\frac{1}{2})^4=1-\frac{1}{16}= \frac{15}{16}

GMATYoda · Answer

Solution: C. 
The only way to not receive at least one of each would be to have all five of the same kind.

That probability is (1/2) * 5, or 1/32.

Because there are two ways to get all five of one kind (all heads or all tails), multiply that by 2 to get 1/16.

All of the other possibilities will result in at least one head and one tail, so the correct answer is 1 - 1/16 = 15/16.

ScottTargetTestPrep · Answer

If a child flips a coin five times in a row, what is the probability that she will receive at least one head and one tail?

A) 3/4
 B) 11/12
 C) 15/16
 D) 31/32
 E) 63/64 /quote]

Since each coin has two faces, head and tail, there are 2^5 = 32 different combinations when flipping a coin five times in a row. Of these 32 combinations, only two of them do not have at least one head and one tail. It occurs when all 5 flips turn out to be heads (HHHHH) or all 5 turn out the be tails (TTTTT). The other 30 combinations will have at least one head and one tail, so the probability is 30/32 = 15/16.

Answer: C

mskx · Answer

Probability that the child flips 5x head:  (\frac{1}{2})^{5} = (\frac{1}{32}) 
Probability that the child flips 5x tail:  (\frac{1}{2})^{5} = (\frac{1}{32})

Probability that the child flips 5x heads or trails  (\frac{1}{2})^{5}  +  (\frac{1}{2})^{5}  =  (\frac{1}{16})

1-(\frac{1}{16}) = (\frac{15}{16})

awssc · Answer

I still don't understand why multiplication should not be used.

Probability of At Least One Head = 1 - Probability of No Heads = 1 - (1/32) = 31/32
Probability of At Least One Tail = 1 - Probability of No Tails = 1 - (1/32) = 31/32

Probability of At Least One Tail AND One Head = (31/32) * (31/32)...

I'm just confused because if I were to consider the Probability of At Least One Tail (which includes the scenario of all tails and no heads) OR the Probability of At Least One Head (which includes the scenario of all heads and no tails), then I'm not finding the probability of at least one head and one tail combination...

If a child flips a coin five times in a row, what is the probability

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If a child flips a coin five times in a row, what is the probability

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