How to calculate what is probability of getting same result 8 times in a row, when flipping coin 1000 times?

Paweł Kiełb :

I've tried to use this code:

function calc (n, c) {
  let a = 0
  const omega = Math.pow(2, n)

  let search1 = ''
  let search2 = ''

  for (let i = 0; i < c; i++) {
    search1 += '0'
  }
  for (let i = 0; i < c; i++) {
    search2 += '1'
  }

  for (let i = 0; i < omega; i++) {
    if (i.toString(2).includes(search1) || i.toString(2).includes(search2)) {
      a++
    }
  }

  const prob = a * 100 / omega
  console.log({ a: a, omega: omega, prob: prob.toFixed(2) })
}

calc(1000, 8)

Which works, but is slow when it comes to big numbers. How can I optimize my code to make it faster? Or maybe there exists a Mathematical solution, that doesn't require to code at all? I just want to know the solution for this problem.

JasonY :

First a Monte Carlo simulation answer: You can find a confidence interval for this simulation by doing some statistical inference on the Bernoulli distribution which I won't do here.

function doesItHappen(l,r){
    var lastValue = null;
    var lastN = 0;
    for(var i = 0; i < l; i++){
        var currentValue = Math.random() > 0.5 ? 1 : 0;
        if(lastValue === currentValue) {
            lastN++;
        } else {
            lastValue = currentValue;
            lastN = 1;
        }
        if(lastN === r) return true;
    }
    return false;
}

function rep(n,l,r){
    var t = 0;
    for(var i = 0; i < n; i++) {
        if(doesItHappen(l,r)) t++;
    }
    return t/n;
}
console.log(rep(100000,1000,8))

Finally the actual Mathematical answer I couldn't find a solution to this question online so I came up with my own method to calculate this in o(n) time and space complexity, you can even get it down to o(1) space complexity by discarding valueStore objects older than the length of consecutive sequence you want. The key thing is to recognise you have to computer all the combinations prior to the current length, similar to a Fibonacci sequence.

function calculateProbability(l,r) {
  var valueStore = [
    { // Initialize it
      totalNumberOfCombinations: 2,
      numberOfCombinationsWithSequence: 0
    }
  ];
  function getValues(index) {
    // combinations with the sequence in it
      // There are two ways a consecutive sequence of r length can occur, it either occured in the previous combination and we flipped a new heads or tails(doesn't matter)
      // Or this flip resulted in a new consecutive sequence of r length occuring (let's say theres k combinations of this)
      // Heres the genius, k must end in a sequence of heads or tails so theres 2 possible endings, the beginnings of k cannot contain the sequence of r consecutive flips
      // If this previous combination ends in a head then the new sequence is all tails and vice versa
      // So k = the combinations of flips without the consective flips before the current sequence
      // k = the totalNumberOfCombinations 8 flips ago - numberOfCombinationsWithSequence 8 flips ago
    if (index === r - 1) {
      // All heads or all tails
      var numberOfCombinationsWithSequence = 2;
    } else if(index < r) {
      var numberOfCombinationsWithSequence = 0;
    } else {
      var numberOfCombinationsWithSequence = valueStore[index - 1].numberOfCombinationsWithSequence * 2 + (valueStore[index - r].totalNumberOfCombinations - valueStore[index - r].numberOfCombinationsWithSequence)
    }
    return {
      // total possible combinations
      // this is just the previous number of combinations but times 2 since we flip again
      totalNumberOfCombinations: valueStore[index - 1].totalNumberOfCombinations * 2,
      numberOfCombinationsWithSequence: numberOfCombinationsWithSequence
    }
  }

  for(var i = 1; i < l; i++) {
    var values = getValues(i);
    valueStore.push(values);
  }

  return valueStore[valueStore.length - 1].numberOfCombinationsWithSequence / valueStore[valueStore.length - 1].totalNumberOfCombinations;
}
console.log(calculateProbability(1000,8));

The 100% accurate answer is 0.9817098435878764 or 98.17%