{"id":2508,"date":"2026-02-02T09:34:40","date_gmt":"2026-02-02T09:34:40","guid":{"rendered":"https:\/\/demo.materiamedica.net\/demo6\/?p=2508"},"modified":"2026-02-02T09:34:40","modified_gmt":"2026-02-02T09:34:40","slug":"chapter-6-binomial-distribution","status":"publish","type":"post","link":"https:\/\/demo.materiamedica.net\/demo6\/chapter-6-binomial-distribution\/","title":{"rendered":"Chapter 6: Binomial Distribution"},"content":{"rendered":"

1. What is the Binomial distribution really?<\/h3>\n

The binomial distribution<\/strong> answers a very simple but extremely common question:<\/p>\n

\n

\u201cIf I repeat the same yes\/no (success\/failure) experiment n<\/strong> times independently, and each trial has the same probability p<\/strong> of success, what is the probability of getting exactly k<\/strong> successes?\u201d<\/p>\n<\/blockquote>\n

That\u2019s it.<\/p>\n

Key ingredients<\/strong> (you must remember these four):<\/p>\n

    \n
  • n<\/strong> = number of independent trials \/ attempts \/ repetitions<\/li>\n
  • p<\/strong> = probability of \u201csuccess\u201d on each<\/strong> trial (0 \u2264 p \u2264 1)<\/li>\n
  • k<\/strong> = number of successes we are interested in (k = 0, 1, 2, …, n)<\/li>\n
  • Each trial must be independent<\/strong> and have exactly two possible outcomes<\/strong> (success \/ failure)<\/li>\n<\/ul>\n

    2. Classic everyday examples<\/h3>\n
    \n
    \n\n\n\n\n\n\n\n\n\n
    Example<\/th>\nn (trials)<\/th>\np (success probability)<\/th>\nWhat k means<\/th>\n<\/tr>\n<\/thead>\n
    Flipping a fair coin 20 times<\/td>\n20<\/td>\n0.5<\/td>\nNumber of heads<\/td>\n<\/tr>\n
    Clicking \u201cbuy now\u201d on a website<\/td>\n1000<\/td>\n0.024<\/td>\nNumber of purchases<\/td>\n<\/tr>\n
    Testing 50 light bulbs<\/td>\n50<\/td>\n0.03<\/td>\nNumber of defective bulbs<\/td>\n<\/tr>\n
    Sending 200 emails in a campaign<\/td>\n200<\/td>\n0.18<\/td>\nNumber of people who open the email<\/td>\n<\/tr>\n
    Shooting 10 free throws<\/td>\n10<\/td>\n0.72<\/td>\nNumber of successful shots<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
    <\/div>\n<\/div>\n<\/div>\n

    3. Generating binomial data in NumPy<\/h3>\n
    \n
    \n
    \n
    Python<\/div>\n
    \n
    # 10,000 simulations of flipping a fair coin 20 times\r\n# \u2192 how many heads each time?\r\nheads = np.random.binomial(n=20, p=0.5, size=10000)\r\n\r\nprint(\"First 15 experiments:\", heads[:15])\r\nprint(\"Average number of heads:\", heads.mean().round(3))\r\nprint(\"Theoretical expected value:\", 20 * 0.5)   # n \u00d7 p<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    4. Visualizing binomial distributions (very important)<\/h3>\n
    \n
    \n
    \n
    Python<\/div>\n
    \n
    fig, axes = plt.subplots(2, 2, figsize=(14, 10), sharey=False)\r\n\r\n# Different combinations of n and p\r\nparams = [\r\n    (10, 0.5, \"Coin flips \u2013 n=10, p=0.5\", \"skyblue\"),\r\n    (40, 0.5, \"n=40, p=0.5\", \"teal\"),\r\n    (30, 0.1, \"Rare events \u2013 n=30, p=0.1\", \"coral\"),\r\n    (30, 0.9, \"Very likely events \u2013 n=30, p=0.9\", \"orchid\")\r\n]\r\n\r\nfor ax, (n, p, title, color) in zip(axes.flat, params):\r\n    data = np.random.binomial(n=n, p=p, size=40000)\r\n    \r\n    sns.histplot(data, bins=np.arange(-0.5, n+1.5, 1), stat=\"probability\",\r\n                 discrete=True, color=color, alpha=0.8, ax=ax)\r\n    \r\n    ax.set_title(title, fontsize=13, pad=12)\r\n    ax.set_xlabel(\"Number of successes (k)\", fontsize=11)\r\n    ax.set_ylabel(\"Probability\", fontsize=11)\r\n    ax.set_xticks(range(0, n+1, max(1, n\/\/10)))\r\n\r\nplt.tight_layout()\r\nplt.show()<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    What you should notice<\/strong>:<\/p>\n
      \n
    • When p = 0.5<\/strong> \u2192 symmetric bell shape<\/li>\n
    • When p is small<\/strong> (e.g. 0.1) \u2192 skewed right (most values near 0)<\/li>\n
    • When p is large<\/strong> (e.g. 0.9) \u2192 skewed left<\/li>\n
    • As n<\/strong> increases \u2192 shape becomes more symmetric and bell-like (\u2192 approaches normal!)<\/li>\n<\/ul>\n
      5. Expected value & variance \u2013 very important formulas<\/h3>\n
      Expected number of successes<\/strong> (mean): E[k] = n \u00d7 p<\/strong><\/p>\n
      Variance<\/strong>: Var(k) = n \u00d7 p \u00d7 (1-p)<\/strong><\/p>\n
      Standard deviation<\/strong>: \u03c3 = \u221a(n \u00d7 p \u00d7 (1-p))<\/strong><\/p>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      n, p = 200, 0.04\r\n\r\nprint(f\"Expected conversions: {n * p:.1f}\")\r\nprint(f\"Standard deviation:   {np.sqrt(n * p * (1-p)):.2f}\")<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \u2192 Most of the time you will see roughly 8 \u00b1 3<\/strong> conversions (mean \u00b1 1 sd)<\/p>\n
      6. Realistic examples you will actually use<\/h3>\n
      Example 1 \u2013 A\/B test simulation<\/strong><\/p>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      # Control group: 12,000 visitors, 3.2% conversion rate\r\ncontrol = np.random.binomial(n=12000, p=0.032, size=10000)\r\n\r\n# Variant B: 12,000 visitors, suppose true rate is 3.8%\r\nvariant = np.random.binomial(n=12000, p=0.038, size=10000)\r\n\r\ndiff = variant - control\r\n\r\nplt.hist(diff, bins=60, color=\"purple\", alpha=0.7)\r\nplt.axvline(0, color='red', linestyle='--', lw=2)\r\nplt.title(\"Difference in conversions (Variant B \u2013 Control)\\n10,000 simulated A\/B tests\")\r\nplt.xlabel(\"Extra conversions from Variant B\")\r\nplt.show()<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      Example 2 \u2013 Quality control<\/strong><\/p>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      # Batch of 500 products, historical defect rate 1.8%\r\ndefects = np.random.binomial(500, 0.018, size=2000)\r\n\r\nprint(\"Probability of \u2265 15 defects:\", np.mean(defects >= 15).round(4))<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      Example 3 \u2013 Email campaign planning<\/strong><\/p>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      sent = 25000\r\nopen_rate = 0.22\r\n\r\nopens = np.random.binomial(sent, open_rate, 5000)\r\n\r\nprint(f\"95% of campaigns will get between {np.percentile(opens, 2.5):.0f} and {np.percentile(opens, 97.5):.0f} opens\")<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      Summary \u2013 Binomial Distribution Cheat Sheet<\/h3>\n
      \n
      \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
      Property<\/th>\nValue \/ Formula<\/th>\n<\/tr>\n<\/thead>\n
      Number of trials<\/td>\nn<\/strong> (fixed)<\/td>\n<\/tr>\n
      Success probability<\/td>\np<\/strong> (same for every trial)<\/td>\n<\/tr>\n
      Possible outcomes<\/td>\nk = 0, 1, 2, …, n<\/td>\n<\/tr>\n
      Expected value (mean)<\/td>\nn \u00d7 p<\/strong><\/td>\n<\/tr>\n
      Variance<\/td>\nn \u00d7 p \u00d7 (1-p)<\/strong><\/td>\n<\/tr>\n
      Standard deviation<\/td>\n\u221a(n \u00d7 p \u00d7 (1-p))<\/strong><\/td>\n<\/tr>\n
      NumPy function<\/td>\nnp.random.binomial(n, p, size=…)<\/td>\n<\/tr>\n
      Shape when p \u2248 0.5<\/td>\nSymmetric (bell-like for large n)<\/td>\n<\/tr>\n
      Shape when p << 0.5<\/td>\nRight-skewed<\/td>\n<\/tr>\n
      Shape when p >> 0.5<\/td>\nLeft-skewed<\/td>\n<\/tr>\n
      Approximation for large n<\/td>\nNormal distribution (Central Limit Theorem)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
      <\/div>\n<\/div>\n<\/div>\n
      Final teacher messages<\/h3>\n
        \n
      1. Whenever you have \u201cnumber of successes in fixed number of yes\/no trials\u201d<\/strong> \u2192 think binomial.<\/li>\n
      2. When n is large and p is not too close to 0 or 1<\/strong> \u2192 binomial looks very much like normal \u2192 you can often use normal approximation.<\/li>\n
      3. Binomial + Poisson connection<\/strong> \u2014 when n is very large and p is very small (n\u00d7p = \u03bb fixed) \u2192 binomial \u2248 Poisson.<\/li>\n<\/ol>\n
        Would you like to continue with any of these next?<\/p>\n
          \n
        • How binomial becomes Poisson (rare events limit)<\/li>\n
        • How binomial becomes normal (large n)<\/li>\n
        • Binomial confidence intervals (real A\/B testing)<\/li>\n
        • Comparing binomial simulations vs theoretical probabilities<\/li>\n
        • Realistic mini-project: simulate A\/B test + power analysis<\/li>\n<\/ul>\n
          Just tell me what feels most interesting or useful right now! \ud83d\ude0a<\/p>\n","protected":false},"excerpt":{"rendered":"
          1. What is the Binomial distribution really? The binomial distribution answers a very simple but extremely common question: \u201cIf I repeat the same yes\/no (success\/failure) experiment n times independently, and each trial has the...<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[75],"tags":[],"class_list":["post-2508","post","type-post","status-publish","format-standard","hentry","category-numpy"],"_links":{"self":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts\/2508","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/comments?post=2508"}],"version-history":[{"count":1,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts\/2508\/revisions"}],"predecessor-version":[{"id":2509,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts\/2508\/revisions\/2509"}],"wp:attachment":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/media?parent=2508"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/categories?post=2508"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/tags?post=2508"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}