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Kate and David each have $10. Together they flip a coin 5

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Kate and David each have $10. Together they flip a coin 5 [#permalink]

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New post Updated on: 13 Apr 2012, 13:52
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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. \(\frac{11}{16}\)

Originally posted by bibha on 13 Jul 2010, 09:43.
Last edited by Bunuel on 13 Apr 2012, 13:52, edited 2 times in total.
Edited the question and added the OA
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Re: probability [#permalink]

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New post 13 Jul 2010, 11:04
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bibha 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. 5/16
b. 15/32
c. 1/2
d. 21/32
e. 11/16


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?".

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

Answer: B.

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

Check this links for similar problems:
viewtopic.php?f=140&t=96468&p=742767&hilit=+google#p742767
viewtopic.php?f=140&t=56812&hilit=+probability+occurring+times
viewtopic.php?f=140&t=88069&hilit=+probability+occurring+times
viewtopic.php?f=140&t=87673&hilit=+probability+occurring+times

Also you can check Probability chapter of Math Book for more (link in my signature).

Hope it helps.
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Re: probability [#permalink]

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New post 04 Jan 2011, 21:32
Thanks for the explanation
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Re: coin probability [#permalink]

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New post 09 Mar 2011, 22:45
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Re: coin probability [#permalink]

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New post 10 Mar 2011, 00:49
Harriet has won, or that 3 <= # of head <=4 because only in those cases Harriet will have Money > 10 and < 15

So Prpb = 5C3(1/2)^3* (1/2)^2 + 5C4(1/2)^4* (1/2)^1

= (1/2)^5{(5 * 4)/2 + 5} = 15 * (1/2)^5 = 15/32

Answer is D.
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Kate and David each have $10. Together they flip a coin 5 [#permalink]

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New post 10 Jan 2015, 00:28
Two options for Kate to get >10 and <15 dollars

Lose (times): 1, 2

Win (times): 4, 3

First option: 1/2 (win)*1/2 (lose)*1/2 (lose)*1/2 (lose)*1/2 (lose)=1/32.

we have 5!/4!*1!=5 such cases. So, (1/32)*5=5/32

Second option : 1/2 (lose)*1/2 (lose)*1/2 (win)*1/2 (win)*1/2 (win)=1/32.

we have 5!/3!*2!=10 such cases. So, (1/32)*10=10/32

5/32+10/32=15/32

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Re: Kate and David each have $10. Together they flip a coin 5 [#permalink]

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New post 07 Jun 2016, 09:04
Temurkhon wrote:
First option: 1/2 (win)*1/2 (lose)*1/2 (lose)*1/2 (lose)*1/2 (lose)=1/32.

we have 5!/4!*1!=5 such cases. So, (1/32)*5=5/32

B


Could anyone please shed me some light on this part? Why is it that we have to count only the number of cases we can arrange winnings and not just 5! ? Kindly thanks!
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Re: Kate and David each have $10. Together they flip a coin 5 [#permalink]

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New post 08 Jun 2016, 02:52
fantaisie wrote:
Temurkhon wrote:
First option: 1/2 (win)*1/2 (lose)*1/2 (lose)*1/2 (lose)*1/2 (lose)=1/32.

we have 5!/4!*1!=5 such cases. So, (1/32)*5=5/32

B


Could anyone please shed me some light on this part? Why is it that we have to count only the number of cases we can arrange winnings and not just 5! ? Kindly thanks!


We are arranging 1 win and 4 loses (WLLL). WLLL can be arranged in 5!/4! number of ways.
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Re: Kate and David each have $10. Together they flip a coin 5 [#permalink]

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New post 15 Feb 2018, 13:27
Hi All,

This is a layered probability question that requires some additional math (the combination formula, or the equivalent "mapping" of all possibilities).

We're told that the coin will be flipped 5 times; with every "win", Kate gets $1 and with every "loss", Kate loses $1. Kate starts with $10. We're asked for the probability that Kate ends up with MORE than $10 but less than $15 after 5 tosses.

Let's start with the total number of possible outcomes. Since each coin has 2 options (heads or tails), there are 2^5 = 32 possible outcomes for the 5 flips (which will include a certain number of similar outcomes in different orders - for example HHTTT and THTHT).

To end up with MORE than $10, Kate has to win MORE tosses than she loses. However, if she were to win all 5 tosses, she'd have $15 and we want her to end up with LESS than $15. This means that Kate has to win EITHER 3 times or 4 times.

Since it does not matter which of the 5 tosses is won, as long as it's either 3 or 4 of them, we can use the combination formula:

Combinations = N!/[K!(N-K)!]

For 3 wins, we have 5!/3!2! = 10 possible combinations of 3 wins

For 4 wins, we have 5!/4!1! = 5 possible combinations of 4 wins

So, there are a total of 10+5 = 15 combinations of 5 tosses that "fit" what we're looking for and 32 possible outcomes total.

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Re: Kate and David each have $10. Together they flip a coin 5   [#permalink] 15 Feb 2018, 13:27
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