diff --git a/core/pom.xml b/core/pom.xml
index bab50f5ce2888..6cb58dbd291c4 100644
--- a/core/pom.xml
+++ b/core/pom.xml
@@ -67,6 +67,10 @@
       <groupId>org.apache.commons</groupId>
       <artifactId>commons-lang3</artifactId>
     </dependency>
+    <dependency>
+      <groupId>org.apache.commons</groupId>
+      <artifactId>commons-math3</artifactId>
+    </dependency>
     <dependency>
       <groupId>com.google.code.findbugs</groupId>
       <artifactId>jsr305</artifactId>
diff --git a/core/src/main/scala/org/apache/spark/rdd/RDD.scala b/core/src/main/scala/org/apache/spark/rdd/RDD.scala
index aa03e9276fb34..2fdf45a0c8b8e 100644
--- a/core/src/main/scala/org/apache/spark/rdd/RDD.scala
+++ b/core/src/main/scala/org/apache/spark/rdd/RDD.scala
@@ -379,8 +379,17 @@ abstract class RDD[T: ClassTag](
     }.toArray
   }
 
-  def takeSample(withReplacement: Boolean, num: Int, seed: Long = Utils.random.nextLong): Array[T] =
-  {
+  /**
+   * Return a fixed-size sampled subset of this RDD in an array
+   *
+   * @param withReplacement whether sampling is done with replacement
+   * @param num size of the returned sample
+   * @param seed seed for the random number generator
+   * @return sample of specified size in an array
+   */
+  def takeSample(withReplacement: Boolean,
+                 num: Int,
+                 seed: Long = Utils.random.nextLong): Array[T] = {
     var fraction = 0.0
     var total = 0
     val multiplier = 3.0
@@ -402,10 +411,11 @@ abstract class RDD[T: ClassTag](
     }
 
     if (num > initialCount && !withReplacement) {
+      // special case not covered in computeFraction
       total = maxSelected
       fraction = multiplier * (maxSelected + 1) / initialCount
     } else {
-      fraction = multiplier * (num + 1) / initialCount
+      fraction = computeFraction(num, initialCount, withReplacement)
       total = num
     }
 
@@ -421,6 +431,22 @@ abstract class RDD[T: ClassTag](
     Utils.randomizeInPlace(samples, rand).take(total)
   }
 
+  private[spark] def computeFraction(num: Int, total: Long, withReplacement: Boolean) : Double = {
+    val fraction = num.toDouble / total
+    if (withReplacement) {
+      var numStDev = 5
+      if (num < 12) {
+        // special case to guarantee sample size for small s
+        numStDev = 9
+      }
+      fraction + numStDev * math.sqrt(fraction / total)
+    } else {
+      val delta = 0.00005
+      val gamma = - math.log(delta)/total
+      math.min(1, fraction + gamma + math.sqrt(gamma * gamma + 2 * gamma * fraction))
+    }
+  }
+
   /**
    * Return the union of this RDD and another one. Any identical elements will appear multiple
    * times (use `.distinct()` to eliminate them).
diff --git a/core/src/main/scala/org/apache/spark/util/random/RandomSampler.scala b/core/src/main/scala/org/apache/spark/util/random/RandomSampler.scala
index 4dc8ada00a3e8..e53103755b279 100644
--- a/core/src/main/scala/org/apache/spark/util/random/RandomSampler.scala
+++ b/core/src/main/scala/org/apache/spark/util/random/RandomSampler.scala
@@ -70,7 +70,7 @@ class BernoulliSampler[T](lb: Double, ub: Double, complement: Boolean = false)
   }
 
   /**
-   *  Return a sampler with is the complement of the range specified of the current sampler.
+   *  Return a sampler which is the complement of the range specified of the current sampler.
    */
   def cloneComplement():  BernoulliSampler[T] = new BernoulliSampler[T](lb, ub, !complement)
 
diff --git a/core/src/test/scala/org/apache/spark/rdd/RDDSuite.scala b/core/src/test/scala/org/apache/spark/rdd/RDDSuite.scala
index e686068f7a99a..5bdcb9bef6d62 100644
--- a/core/src/test/scala/org/apache/spark/rdd/RDDSuite.scala
+++ b/core/src/test/scala/org/apache/spark/rdd/RDDSuite.scala
@@ -22,6 +22,7 @@ import scala.reflect.ClassTag
 
 import org.scalatest.FunSuite
 
+import org.apache.commons.math3.distribution.PoissonDistribution
 import org.apache.spark._
 import org.apache.spark.SparkContext._
 import org.apache.spark.rdd._
@@ -494,56 +495,84 @@ class RDDSuite extends FunSuite with SharedSparkContext {
     assert(sortedTopK === nums.sorted(ord).take(5))
   }
 
+  test("computeFraction") {
+    // test that the computed fraction guarantees enough datapoints in the sample with a failure rate <= 0.0001
+    val data = new EmptyRDD[Int](sc)
+    val n = 100000
+
+    for (s <- 1 to 15) {
+      val frac = data.computeFraction(s, n, true)
+      val qpois = new PoissonDistribution(frac * n)
+      assert(qpois.inverseCumulativeProbability(0.0001) >= s, "Computed fraction is too low")
+    }
+    for (s <- 1 to 15) {
+      val frac = data.computeFraction(s, n, false)
+      val qpois = new PoissonDistribution(frac * n)
+      assert(qpois.inverseCumulativeProbability(0.0001) >= s, "Computed fraction is too low")
+    }
+    for (s <- List(1, 10, 100, 1000)) {
+      val frac = data.computeFraction(s, n, true)
+      val qpois = new PoissonDistribution(frac * n)
+      assert(qpois.inverseCumulativeProbability(0.0001) >= s, "Computed fraction is too low")
+    }
+    for (s <- List(1, 10, 100, 1000)) {
+      val frac = data.computeFraction(s, n, false)
+      val qpois = new PoissonDistribution(frac * n)
+      assert(qpois.inverseCumulativeProbability(0.0001) >= s, "Computed fraction is too low")
+    }
+  }
+
   test("takeSample") {
-    val data = sc.parallelize(1 to 100, 2)
+    val n = 1000000
+    val data = sc.parallelize(1 to n, 2)
     
     for (num <- List(5, 20, 100)) {
       val sample = data.takeSample(withReplacement=false, num=num)
       assert(sample.size === num)        // Got exactly num elements
       assert(sample.toSet.size === num)  // Elements are distinct
-      assert(sample.forall(x => 1 <= x && x <= 100), "elements not in [1, 100]")
+      assert(sample.forall(x => 1 <= x && x <= n), "elements not in [1, 100]")
     }
     for (seed <- 1 to 5) {
       val sample = data.takeSample(withReplacement=false, 20, seed)
       assert(sample.size === 20)        // Got exactly 20 elements
       assert(sample.toSet.size === 20)  // Elements are distinct
-      assert(sample.forall(x => 1 <= x && x <= 100), "elements not in [1, 100]")
+      assert(sample.forall(x => 1 <= x && x <= n), "elements not in [1, 100]")
     }
     for (seed <- 1 to 5) {
-      val sample = data.takeSample(withReplacement=false, 200, seed)
+      val sample = data.takeSample(withReplacement=false, 100, seed)
       assert(sample.size === 100)        // Got only 100 elements
       assert(sample.toSet.size === 100)  // Elements are distinct
-      assert(sample.forall(x => 1 <= x && x <= 100), "elements not in [1, 100]")
+      assert(sample.forall(x => 1 <= x && x <= n), "elements not in [1, 100]")
     }
     for (seed <- 1 to 5) {
       val sample = data.takeSample(withReplacement=true, 20, seed)
       assert(sample.size === 20)        // Got exactly 20 elements
-      assert(sample.forall(x => 1 <= x && x <= 100), "elements not in [1, 100]")
+      assert(sample.forall(x => 1 <= x && x <= n), "elements not in [1, 100]")
     }
     {
       val sample = data.takeSample(withReplacement=true, num=20)
       assert(sample.size === 20)        // Got exactly 100 elements
       assert(sample.toSet.size <= 20, "sampling with replacement returned all distinct elements")
-      assert(sample.forall(x => 1 <= x && x <= 100), "elements not in [1, 100]")
+      assert(sample.forall(x => 1 <= x && x <= n), "elements not in [1, 100]")
     }
     {
-      val sample = data.takeSample(withReplacement=true, num=100)
-      assert(sample.size === 100)        // Got exactly 100 elements
+      val sample = data.takeSample(withReplacement=true, num=n)
+      assert(sample.size === n)        // Got exactly 100 elements
       // Chance of getting all distinct elements is astronomically low, so test we got < 100
-      assert(sample.toSet.size < 100, "sampling with replacement returned all distinct elements")
-      assert(sample.forall(x => 1 <= x && x <= 100), "elements not in [1, 100]")
+      assert(sample.toSet.size < n, "sampling with replacement returned all distinct elements")
+      assert(sample.forall(x => 1 <= x && x <= n), "elements not in [1, 100]")
     }
     for (seed <- 1 to 5) {
-      val sample = data.takeSample(withReplacement=true, 100, seed)
-      assert(sample.size === 100)        // Got exactly 100 elements
+      val sample = data.takeSample(withReplacement=true, n, seed)
+      assert(sample.size === n)        // Got exactly 100 elements
       // Chance of getting all distinct elements is astronomically low, so test we got < 100
-      assert(sample.toSet.size < 100, "sampling with replacement returned all distinct elements")
+      assert(sample.toSet.size < n, "sampling with replacement returned all distinct elements")
     }
     for (seed <- 1 to 5) {
-      val sample = data.takeSample(withReplacement=true, 200, seed)
-      assert(sample.size === 200)        // Got exactly 200 elements
+      val sample = data.takeSample(withReplacement=true, 2*n, seed)
+      assert(sample.size === 2*n)        // Got exactly 200 elements
       // Chance of getting all distinct elements is still quite low, so test we got < 100
-      assert(sample.toSet.size < 100, "sampling with replacement returned all distinct elements")
+      assert(sample.toSet.size < n, "sampling with replacement returned all distinct elements")
     }
   }
 
diff --git a/pom.xml b/pom.xml
index 7bf9f135fd340..01d6eef32be63 100644
--- a/pom.xml
+++ b/pom.xml
@@ -245,6 +245,11 @@
 	    <artifactId>commons-codec</artifactId>
 	    <version>1.5</version>
       </dependency>
+      <dependency>
+        <groupId>org.apache.commons</groupId>
+        <artifactId>commons-math3</artifactId>
+        <version>3.2</version>
+      </dependency>
       <dependency>
         <groupId>com.google.code.findbugs</groupId>
         <artifactId>jsr305</artifactId>
diff --git a/project/SparkBuild.scala b/project/SparkBuild.scala
index 8ef1e91f609fb..a6b6c26a49395 100644
--- a/project/SparkBuild.scala
+++ b/project/SparkBuild.scala
@@ -331,6 +331,7 @@ object SparkBuild extends Build {
     libraryDependencies ++= Seq(
         "com.google.guava"           % "guava"            % "14.0.1",
         "org.apache.commons"         % "commons-lang3"    % "3.3.2",
+        "org.apache.commons"         % "commons-math3"    % "3.2",
         "com.google.code.findbugs"   % "jsr305"           % "1.3.9",
         "log4j"                      % "log4j"            % "1.2.17",
         "org.slf4j"                  % "slf4j-api"        % slf4jVersion,
diff --git a/python/pyspark/rdd.py b/python/pyspark/rdd.py
index 07578b8d937fc..b400404ad97c7 100644
--- a/python/pyspark/rdd.py
+++ b/python/pyspark/rdd.py
@@ -31,6 +31,7 @@
 import warnings
 import heapq
 from random import Random
+from math import sqrt, log, min
 
 from pyspark.serializers import NoOpSerializer, CartesianDeserializer, \
     BatchedSerializer, CloudPickleSerializer, PairDeserializer, pack_long
@@ -374,7 +375,7 @@ def takeSample(self, withReplacement, num, seed=None):
             total = maxSelected
             fraction = multiplier * (maxSelected + 1) / initialCount
         else:
-            fraction = multiplier * (num + 1) / initialCount
+            fraction = self._computeFraction(num, initialCount, withReplacement)
             total = num
 
         samples = self.sample(withReplacement, fraction, seed).collect()
@@ -390,6 +391,18 @@ def takeSample(self, withReplacement, num, seed=None):
         sampler.shuffle(samples)
         return samples[0:total]
 
+    def _computeFraction(self, num, total, withReplacement):
+        fraction = float(num)/total
+        if withReplacement:
+            numStDev = 5
+            if (num < 12):
+                numStDev = 9
+            return fraction + numStDev * sqrt(fraction/total)
+        else:
+            delta = 0.00005
+            gamma = - log(delta)/total
+            return min(1, fraction + gamma + sqrt(gamma * gamma + 2* gamma * fraction))
+
     def union(self, other):
         """
         Return the union of this RDD and another one.