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# m05 Q17 Probability/Combination

Author Message
Intern
Joined: 04 Jul 2013
Posts: 17
Location: India
Concentration: Operations, Technology
WE: Operations (Manufacturing)
Followers: 2

Kudos [?]: 33 [0], given: 7

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01 Aug 2013, 10:07
The answer is 15/32 which comes when we consider the two cases inwhich Kate can actually get a total of more than 10 and less than 15...these are when Kate gets 3 tosses in her favour and 2 against her ie a total of 11...similarly when she gets 4 in her favour and 1 against ie total of 13....
Note a total of 12 and 14 is not possible....so a total of 15 possibilities of 32 are there....

Thanks....

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Joined: 02 Sep 2009
Posts: 36520
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Kudos [?]: 92918 [0], given: 10528

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11 Aug 2013, 01:08
d2touge wrote:
Kate and David each have $10. Together they flip a coin 5 times. Every time the coin lands on heads, Kate gives David$1. Every time the coin lands on tails, David gives Kate $1. After the coin is flipped 5 times, what is the probability that Kate has more than$10 but less than $15. (A) $$\frac{5}{16}$$ (B) $$\frac{15}{32}$$ (C) $$\frac{1}{2}$$ (D) $$\frac{21}{32}$$ (E) $$\frac{11}{16}$$ [Reveal] Spoiler: OA B Source: GMAT Club Tests - hardest GMAT questions So to answer this, we find the total combinations when you flip the coin 5 times. --> 2^5 = 32. The explanation tells us to find the combinations when kate wins 3 times and 4 times. --> 5C3 and 5C4. we get 10 and 5 respectively. Lastly, we simply find the probability (5/32) + (10/32) = 15/32. and that is the answer.. ***Why do we only find the combinations when Kate wins 3 times and 4 times? Why not find 1 and 2 times? After 5 tries Kate to have more than initial sum of 10$ and less than 15$must win 3 or 4 times (if she wins 2 or less times she'll have less than 10$ and if she wins 5 times she'll have 15\$).

So the question becomes "what is the probability of getting 3 or 4 tails in 5 tries?".

$$P(t=3 \ or \ t=4)=P(t=3)+P(t=4)=C^3_5*(\frac{1}{2})^5+C^4_5*(\frac{1}{2})^5=\frac{15}{32}$$

To elaborate more:

If the probability of a certain event is $$p$$, then the probability of it occurring $$k$$ times in $$n$$-time sequence is: $$P = C^k_n*p^k*(1-p)^{n-k}$$

For example for the case of getting 3 tails in 5 tries:
$$n=5$$ (5 tries);
$$k=3$$ (we want 3 tail);
$$p=\frac{1}{2}$$ (probability of tail is 1/2).

So, $$P = C^k_n*p^k*(1-p)^{n-k}=C^3_5*(\frac{1}{2})^3*(1-\frac{1}{2})^{(5-3)}=C^3_5*(\frac{1}{2})^5$$

OR: probability of scenario t-t-t-h-h is $$(\frac{1}{2})^3*(\frac{1}{2})^2$$, but t-t-t-h-h can occur in different ways:

t-t-t-h-h - first three tails and fourth and fifth heads;
h-h-t-t-t - first two heads and last three tails;
t-h-h-t-t - first tail, then two heads, then two tails;
...

Certain # of combinations. How many combinations are there? Basically we are looking at # of permutations of five letters t-t-t-h-h, which is $$\frac{5!}{3!2!}$$.

Hence $$P=\frac{5!}{3!2!}*(\frac{1}{2})^5$$.

Hope it helps.
_________________
Re: m05 Q17 Probability/Combination   [#permalink] 11 Aug 2013, 01:08

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# m05 Q17 Probability/Combination

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