db.foo.drop();

for (let i = 0; i < 1000; i++) {
    db.foo.insert({a: i});
    db.foo.insert({a: 'X'});
}
db.foo.createIndex({a: 1});
db.foo.find({a: { '$type': 'string' } } ).toArray().length;
db.foo.find({a: { '$type': 'string' } } ).explain().queryPlanner.winningPlan;

db.foo.find({a: { '$type': 'int' } } ).toArray().length;
db.foo.find({a: { '$type': 'int' } } ).explain().queryPlanner.winningPlan;

{
  isCached: false,
  stage: 'FETCH',
  costEstimate: 0.4352537,
  cardinalityEstimate: 0,
  estimatesMetadata: { ceSource: 'Sampling' },
  inputStage: {
    stage: 'IXSCAN',
    costEstimate: 0.427765,
    cardinalityEstimate: 0, <<<=== WRONG
    numKeysEstimate: 1000,
    estimatesMetadata: { ceSource: 'Sampling' },
    filter: { a: { '$type': [ 16 ] } },
    keyPattern: { a: 1 },
    indexName: 'a_1',
    isMultiKey: false,
    multiKeyPaths: { a: [] },
    isUnique: false,
    isSparse: false,
    isPartial: false,
    indexVersion: 2,
    direction: 'forward',
    indexBounds: { a: [ '[nan, inf]' ] }
  }
}