What is the probability that 10 coins selected randomly from 100 coins add up to $1?
Assume the 100 coins consist randomly of pennies, nickels, dimes, and quarters. Each coin has an equal probability of being selected.
Just as a side note, I got the idea for this question at toll booth. So I suppose this does have some type of real-world application.
There are several ways to make $1 out of 10 coins:
A
quaters: 3
dimes 2
penies 5
B
quateres 2
dimes 2
nickels 6
C
quaters 1
dimes 6
nickels 3
D
qaters 0
nickels 10
The number of ways to construct selections is
nA: 10!/(3!2!5!) = 2520
nB: 10!/2!2!6! = 1260
nC: 10!/1!6!3! = 840
nD: 10!/10! = 1
Total volume of desirable configuration space is
nA+nB+nC+nD = 4621 out of
volume of entire configuration space 4^10
Answer:
4621/4^10 = 0.440 692 901 611 328 125 %
(decimal fraction above is exact)
May 29th, 2010 at 6:49 am
You would need to know the number of each coin to accurately predict
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May 29th, 2010 at 7:38 am
not enuf info i would say…it doesn’t say how many of each coin there is
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May 29th, 2010 at 7:43 am
If each coin has an equal probability of appearing, that means that, assuming size doesn’t matter, that getting a certain type of coin is 1/4.
penny(1), nickel(2), dime(3), quarter(4). 25 of each, out of 100, 25/100 or 1/4.
If you took 10 coins out, then theoretically, 2.5 of each type of coin will appear.
10coins/4options=2.5per type of coin
This is the hard part. I can’t help you there, but then you’ll have to find all the possible combinations in 10 coins.
4 coins, each coin you pick will not affect the next pick. 4 more coins for each type of coin. 4 * 4=16 (two picks), 4 more after that, 16 * 4=64 (three). Or, 4^10, FOUR to the power of TEN. 40 000 000 000(forty billion). There are 4^10 possible combinations for ten picks from your bag, but I can’t say the exact probability of making the total >= $1. Here, you’re on your own. Sorry.
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May 29th, 2010 at 8:18 am
There are several ways to make $1 out of 10 coins:
A
quaters: 3
dimes 2
penies 5
B
quateres 2
dimes 2
nickels 6
C
quaters 1
dimes 6
nickels 3
D
qaters 0
nickels 10
The number of ways to construct selections is
nA: 10!/(3!2!5!) = 2520
nB: 10!/2!2!6! = 1260
nC: 10!/1!6!3! = 840
nD: 10!/10! = 1
Total volume of desirable configuration space is
nA+nB+nC+nD = 4621 out of
volume of entire configuration space 4^10
Answer:
4621/4^10 = 0.440 692 901 611 328 125 %
(decimal fraction above is exact)
References :
May 29th, 2010 at 8:44 am
1/100000000000000000000 Because you do 1/100 to the tenth power. Because you do 1/100 for one coin and do that ten times. Then the result(1/100000000000000000000) is the probability of all 10 coins adding to be $1.
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May 29th, 2010 at 8:51 am
since the coins have equal probability of being selected (at the start) there are 25 of each.
10D 25C10
1Q6D3N 25*25C6*25C3
2Q2D6N 25C2 ^2 * 25C6
3Q2D5P 25C3 * 25C2 * 25C5
sum these over the sample space which is 100C10
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May 29th, 2010 at 9:17 am
not enough info
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May 29th, 2010 at 10:07 am
I get about 0.44%.
The following code is run in R. It creates 2000 collections of 100 coins using a random multinomial vector with equal probability of penny, nickels, dimes and quarters.
each of the collections goes through 1000 simulations where a sample of 10 coins is selected without replacement and if the sum of the 10 coins is equal to 1.00 then the count variable is indexed.
the probability is estimated as count / (sims * collections)
===
collections <- 10000
sims <- 1000
count <- 0
x <- rmultinom(collections, 100, c(.25, .25, .25, .25))
for(i in 1:collections)
{
t <- x[,i]
set <- c( rep(0.01, t[1]), rep(0.05, t[2]), rep(0.10, t[3]), rep(0.25, t[4]))
for(j in 1:sims)
{
if(sum(sample(set, 10)) == 1.0) count <- count + 1;
}
}
count / (sims * collections)
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May 29th, 2010 at 10:12 am
You would need to know the number of each coin to accurately predict
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May 29th, 2010 at 10:43 am
10/100
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May 29th, 2010 at 10:50 am
Am sure there will be accurate ways to calculate this if you ask math geniuses.
but if you ask nob head like me (which you did
25+10+5+1+25+10+5+1+10+5 will be average pick. which is equal 97cents
)
I’ll give ya a discount… forget the 3 cents and lets move on
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