diff --git a/src/random.cpp b/src/random.cpp
--- a/src/random.cpp
+++ b/src/random.cpp
@@ -11,6 +11,7 @@
 #endif
 #include <crypto/sha512.h>
 #include <logging.h> // for LogPrint()
+#include <support/allocators/secure.h>
 #include <support/cleanse.h>
 #include <sync.h>      // for WAIT_LOCK
 #include <util/time.h> // for GetTime()
@@ -25,8 +26,6 @@
 #include <mutex>
 #include <thread>
 
-#include <support/allocators/secure.h>
-
 #ifndef WIN32
 #include <fcntl.h>
 #include <sys/time.h>
@@ -542,8 +541,10 @@
     ProcRand(nullptr, 0, RNGLevel::SLEEP);
 }
 
+bool g_mock_deterministic_tests{false};
+
 uint64_t GetRand(uint64_t nMax) noexcept {
-    return FastRandomContext().randrange(nMax);
+    return FastRandomContext(g_mock_deterministic_tests).randrange(nMax);
 }
 
 int GetRandInt(int nMax) noexcept {
diff --git a/src/test/bloom_tests.cpp b/src/test/bloom_tests.cpp
--- a/src/test/bloom_tests.cpp
+++ b/src/test/bloom_tests.cpp
@@ -1195,6 +1195,9 @@
 }
 
 BOOST_AUTO_TEST_CASE(rolling_bloom) {
+    SeedInsecureRand(/* deterministic */ true);
+    g_mock_deterministic_tests = true;
+
     // last-100-entry, 1% false positive:
     CRollingBloomFilter rb1(100, 0.01);
 
@@ -1218,13 +1221,8 @@
     for (int i = 0; i < 10000; i++) {
         if (rb1.contains(RandomData())) ++nHits;
     }
-    // Run test_bitcoin with --log_level=message to see BOOST_TEST_MESSAGEs:
-    BOOST_TEST_MESSAGE("RollingBloomFilter got "
-                       << nHits << " false positives (~100 expected)");
-
-    // Insanely unlikely to get a fp count outside this range:
-    BOOST_CHECK(nHits > 25);
-    BOOST_CHECK(nHits < 175);
+    // Expect about 100 hits
+    BOOST_CHECK_EQUAL(nHits, 75);
 
     BOOST_CHECK(rb1.contains(data[DATASIZE - 1]));
     rb1.reset();
@@ -1249,11 +1247,8 @@
     for (int i = 0; i < DATASIZE; i++) {
         if (rb1.contains(data[i])) ++nHits;
     }
-    // Expect about 5 false positives, more than 100 means
-    // something is definitely broken.
-    BOOST_TEST_MESSAGE("RollingBloomFilter got "
-                       << nHits << " false positives (~5 expected)");
-    BOOST_CHECK(nHits < 100);
+    // Expect about 5 false positives
+    BOOST_CHECK_EQUAL(nHits, 6);
 
     // last-1000-entry, 0.01% false positive:
     CRollingBloomFilter rb2(1000, 0.001);
@@ -1264,6 +1259,7 @@
     for (int i = 0; i < DATASIZE; i++) {
         BOOST_CHECK(rb2.contains(data[i]));
     }
+    g_mock_deterministic_tests = false;
 }
 
 BOOST_AUTO_TEST_SUITE_END()
diff --git a/src/test/random_tests.cpp b/src/test/random_tests.cpp
--- a/src/test/random_tests.cpp
+++ b/src/test/random_tests.cpp
@@ -19,9 +19,16 @@
 
 BOOST_AUTO_TEST_CASE(fastrandom_tests) {
     // Check that deterministic FastRandomContexts are deterministic
+    g_mock_deterministic_tests = true;
     FastRandomContext ctx1(true);
     FastRandomContext ctx2(true);
 
+    for (int i = 10; i > 0; --i) {
+        BOOST_CHECK_EQUAL(GetRand(std::numeric_limits<uint64_t>::max()),
+                          uint64_t{10393729187455219830U});
+        BOOST_CHECK_EQUAL(GetRandInt(std::numeric_limits<int>::max()),
+                          int{769702006});
+    }
     BOOST_CHECK_EQUAL(ctx1.rand32(), ctx2.rand32());
     BOOST_CHECK_EQUAL(ctx1.rand32(), ctx2.rand32());
     BOOST_CHECK_EQUAL(ctx1.rand64(), ctx2.rand64());
@@ -36,6 +43,13 @@
     BOOST_CHECK(ctx1.randbytes(50) == ctx2.randbytes(50));
 
     // Check that a nondeterministic ones are not
+    g_mock_deterministic_tests = false;
+    for (int i = 10; i > 0; --i) {
+        BOOST_CHECK(GetRand(std::numeric_limits<uint64_t>::max()) !=
+                    uint64_t{10393729187455219830U});
+        BOOST_CHECK(GetRandInt(std::numeric_limits<int>::max()) !=
+                    int{769702006});
+    }
     {
         FastRandomContext ctx3, ctx4;
         // extremely unlikely to be equal
diff --git a/src/test/test_bitcoin.h b/src/test/test_bitcoin.h
--- a/src/test/test_bitcoin.h
+++ b/src/test/test_bitcoin.h
@@ -31,6 +31,11 @@
  */
 extern FastRandomContext g_insecure_rand_ctx;
 
+/**
+ * Flag to make GetRand in random.h return the same number
+ */
+extern bool g_mock_deterministic_tests;
+
 static inline void SeedInsecureRand(bool deterministic = false) {
     g_insecure_rand_ctx = FastRandomContext(deterministic);
 }