Probabilistic approach puzzle

Question

A few hours earlier, I got this puzzle:

Suppose, you toss a coin n times, then what is the probability of getting m number of heads? ( where m belongs to the set of all prime numbers)

For instance, take n = 2, then
SS = { HH, HT, TH, TT }

So, output is 1/4 (for HH case)

Then, for n = 3,
SS = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT }

So, output is (1 + 3)/8 (for 3 H's case and 2 H's cases respectively)

Input: n ( an integer, 0 <= n <= 1000 )
Output: The probability (in any distinguishable form, i.e., decimal, exponential, fraction, etc)

This is a code-golf, so fewest bytes would win!

Answer 1

Wolfram Language (Mathematica), 33 bytes

Uses Binomial and outputs ,really fast, fractions

Tr@Binomial[#,Prime@Range@#]/2^#&

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The original code according to my formula was

Tr@Binomial[#,Prime@Range@PrimePi@#]/2^#&    

but, as @attinat commented, instead of searching all Primes < n we can
search the first n primes because every prime > n returns zero binomial.
In this way we save 8 bytes

Here is also the graph of the first 1000 cases which looks pretty cool

Answer 2

Jelly, 7 bytes

cÆRSH⁸¡

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How?

cÆRSH⁸¡ - Link@ integer, n                   e.g. 20
 ÆR     - primes to n                             [2,3,5,7,11,13,17,19]
c       - (n) choose (vectorises)                 [190,1140,15504,77520,167960,77520,1140,20]
   S    - sum                                     340994
      ¡ - repeat...
     ⁸  - ...times: chain's left argument, n
    H   - ...action: halve
        -   (170497, 85248.5, ..., ~0.65, ~0.325) 0.3251972198486328

Answer 3

05AB1E, 10 7 bytes

ÅPcOso/

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Explanation

ÅP       # push a list of primes upto and including input
  c      # push choose(input, prime) for each prime
   O     # sum
    so   # push 2^input
      /  # divide 

Answer 4

Octave/MATLAB with Statistics Package/Toolbox, 32 bytes

@(n)sum(binopdf(primes(n),n,.5))

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Answer 5

Python 3, 198 125 97 bytes

lambda n:f(n)/2**n
f=lambda n,h=0:h>1>sum(h%i<1for i in range(2,h))if n<1else f(n-1,h+1)+f(n-1,h)

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To aid in understanding, I first generate all the of the possible combinations, then I get then total heads if that is prime, then I add the total number of primes. The second function, h generates the result from function f. Bellow is the original algorithm before minimizing it.

def f(n,h=""):
  count = 0
  if n==0:
    hCount=h.count("h")
    for i in range(2, hCount):
      if hCount % i == 0:
        return 0
    if hCount < 2:
      return 0
    return 1
  count += f(n-1,h+"h")
  count += f(n-1,h+"t")
  return count

def h(n):
  return f(n), (2**n)

print(h(4))

Credits:

• From 125 to 97 bytes by attinat

Answer 6

Excel Formula, 311 309 bytes

The following should be entered as an array formula (Ctrl+Shift+Enter):

=SUM(IFERROR(COMBIN(A1,SMALL(IF(MMULT(--(IF(ROW(INDIRECT("2:"&A1))>TRANSPOSE(ROW(INDIRECT("2:"&A1))),MOD(ROW(INDIRECT("2:"&A1)),(ROW(INDIRECT("2:"&A1))>TRANSPOSE(ROW(INDIRECT("2:"&A1))))*TRANSPOSE(ROW(INDIRECT("2:"&A1)))))=0),ROW(INDIRECT("2:"&A1)))=0,ROW(INDIRECT("2:"&A1))),ROW(INDIRECT("1:"&A1)))),0))/(2^A1)

Where A1 is the number to test.

Examples:

A1=15
Result: 0.34997558593750
A1=25
Result: 0.341329127550125

Credits

-2 thanks to Sophia Lechner!

Answer 7

Gaia, 12 8 bytes

:ṅK¦Σ@z÷

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:		| dup input n
 ṅ		| push first n prime numbers, [2..k]
  K¦		| push n choose k (0 if k > n)
    Σ		| sum
     @z		| push 2^n
       ÷	| divide

Answer 8

Pari/GP, 42 bytes

n->sum(i=0,n,binomial(n,i)*isprime(i))/2^n

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Answer 9

Retina, 41 bytes

K`
"$+"+%`^
$"H
*\Cm`^(?!(..+)\1+$)..
m`^

Try it online! Output is as a ratio. Explanation:

K`
"$+"+%`^
$"H

Generate all the possible coin tosses of n coins, but keep only the heads. (T$"H would keep the tails as well.)

*\Cm`^(?!(..+)\1+$)..

Count how many are prime and print on a separate line.

m`^

Count how many there are altogether.

Answer 10

JavaScript (ES7), 71 bytes

n=>(s=1,g=i=>i<n&&(s*=(n-i)/++i)*(p=d=>i%--d?p(d):d==1)(i)+g(i))``/2**n

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Answer 11

Jelly, 7 bytes

Ø.ṗ§ẒÆm

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Different approach from Jonathan Allan's answer.

Answer 12

Japt -x, 13 bytes

õÈj *UàX /2pU

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