Code to produce a probability distribution of how likely a gender breakdown in conference speakers is, and also to experiment to produce the result

experiment.rb

# This program will select at random a set of speakers and look at the gender breakdown given 20% women

total_speakers = 15
percentage_of_women = 0.25

hist, list, results = {}, [], []

(1000 * (1 - percentage_of_women)).floor.times{list << 0}
(1000 * percentage_of_women).floor.times{list << 1}

100000.times{results << list.shuffle.take(total_speakers).select{|i|i==1}.count}
results.each{|num_women| hist[num_women] ||= 0; hist[num_women] += 1}

puts "Out of 100000 trials, the number of trials that resulted in X women selected follow."
hist.keys.sort.each{|key| puts "#{key}: #{hist[key]}"}

speaker_prob.rb

total_speakers = 15

def fact(n)
  return n < 2 ? 1 : n * fact(n - 1)
end

probs = []
(total_speakers+1).times do |i|
  num_women = i
  num_men   = total_speakers - i

  # how many ways can this outcome occur?
  num_outcomes = fact(total_speakers) / (fact(num_women) * fact(num_men))

  # what is the probability of this outcome occuring once?
  prob = (8.0/10.0)**(num_men) * (2.0/10.0)**(num_women)

  # what is the total probability of the distribution?
  total_prob = prob * num_outcomes
  simple_percent = (total_prob * 1000).floor.to_f / 10
  probs << simple_percent
  puts "#{num_men} men and #{num_women} women = #{simple_percent}%"
end

rolling = 100
probs.each_with_index do |p,i|
  simple_percent = (rolling * 10).floor.to_f / 10
  puts "Probability of >#{i-1} women: #{simple_percent}%" unless i == 0
  rolling = rolling - p
end

You, sir, are awesome. Thank you for being part of the solution. :)

Output for those not wanting to run it (probability of a given number of female speakers at a conf with 15 speakers given a 20% female selection pool):

Probability of >0 women: 96.5%
Probability of >1 women: 83.4%
Probability of >2 women: 60.4%
Probability of >3 women: 35.4%
Probability of >4 women: 16.7%
Probability of >5 women: 6.4%
Probability of >6 women: 2.2%
Probability of >7 women: 0.9%
Probability of >8 women: 0.6%
Probability of >9 women: 0.6%
Probability of >10 women: 0.6%
Probability of >11 women: 0.6%
Probability of >12 women: 0.6%
Probability of >13 women: 0.6%
Probability of >14 women: 0.6%

Looks like once the probabilities get really low there's some kind of bug (since all after 9 are the same).

Is the number of people with something interesting to talk about great enough to approximate sampling with replacement? (that is, if you select a male speaker for the first slot does it appreciably shift the probabilities for the second slot?)

Out of 100000 trials, the number of trials that resulted in X women selected follow.

0: 1303
1: 6650
2: 15573
3: 22670
4: 22583
5: 16660
6: 9089
7: 3852
8: 1232
9: 309
10: 70
11: 6
12: 3