Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 4 times and lost 6 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Car
6. Car
7. Goat
8. Goat
9. Car
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 308 times and lost 692 times.

1. Car
2. Car
3. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Car
9. Car
10. Goat
11. Goat
12. Car
13. Goat
14. Goat
15. Goat
16. Goat
17. Car
18. Goat
19. Car
20. Car
21. Goat
22. Goat
23. Goat
24. Car
25. Goat
26. Car
27. Car
28. Goat
29. Goat
30. Goat
31. Goat
32. Goat
33. Goat
34. Goat
35. Car
36. Car
37. Goat
38. Car
39. Goat
40. Car
41. Goat
42. Goat
43. Car
44. Goat
45. Goat
46. Car
47. Car
48. Car
49. Car
50. Goat
51. Car
52. Goat
53. Goat
54. Goat
55. Goat
56. Goat
57. Car
58. Goat
59. Car
60. Goat
61. Goat
62. Goat
63. Goat
64. Goat
65. Car
66. Goat
67. Goat
68. Car
69. Car
70. Goat
71. Car
72. Goat
73. Goat
74. Goat
75. Goat
76. Car
77. Goat
78. Goat
79. Goat
80. Car
81. Car
82. Goat
83. Goat
84. Goat
85. Goat
86. Goat
87. Car
88. Goat
89. Goat
90. Goat
91. Car
92. Goat
93. Goat
94. Goat
95. Car
96. Goat
97. Goat
98. Goat
99. Goat
100. Goat
101. Goat
102. Car
103. Goat
104. Goat
105. Goat
106. Goat
107. Goat
108. Goat
109. Goat
110. Car
111. Goat
112. Goat
113. Goat
114. Car
115. Goat
116. Goat
117. Goat
118. Goat
119. Goat
120. Goat
121. Goat
122. Goat
123. Goat
124. Goat
125. Car
126. Goat
127. Goat
128. Goat
129. Car
130. Goat
131. Car
132. Car
133. Goat
134. Car
135. Car
136. Goat
137. Goat
138. Goat
139. Car
140. Goat
141. Goat
142. Goat
143. Goat
144. Goat
145. Goat
146. Car
147. Goat
148. Goat
149. Goat
150. Goat
151. Goat
152. Goat
153. Goat
154. Car
155. Goat
156. Goat
157. Goat
158. Goat
159. Car
160. Car
161. Goat
162. Goat
163. Goat
164. Goat
165. Car
166. Goat
167. Goat
168. Car
169. Goat
170. Car
171. Goat
172. Goat
173. Car
174. Goat
175. Goat
176. Car
177. Goat
178. Goat
179. Car
180. Car
181. Goat
182. Car
183. Goat
184. Car
185. Goat
186. Goat
187. Car
188. Car
189. Car
190. Car
191. Goat
192. Goat
193. Goat
194. Goat
195. Car
196. Goat
197. Car
198. Car
199. Car
200. Goat
201. Car
202. Goat
203. Car
204. Goat
205. Goat
206. Goat
207. Goat
208. Goat
209. Goat
210. Goat
211. Goat
212. Goat
213. Car
214. Goat
215. Goat
216. Goat
217. Goat
218. Goat
219. Car
220. Goat
221. Car
222. Car
223. Car
224. Goat
225. Goat
226. Goat
227. Goat
228. Goat
229. Goat
230. Car
231. Car
232. Goat
233. Goat
234. Goat
235. Goat
236. Car
237. Goat
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Car
244. Goat
245. Goat
246. Goat
247. Goat
248. Goat
249. Goat
250. Goat
251. Car
252. Goat
253. Goat
254. Goat
255. Car
256. Goat
257. Car
258. Goat
259. Goat
260. Car
261. Car
262. Goat
263. Goat
264. Car
265. Car
266. Goat
267. Car
268. Goat
269. Goat
270. Car
271. Goat
272. Car
273. Car
274. Goat
275. Goat
276. Goat
277. Car
278. Goat
279. Goat
280. Car
281. Goat
282. Goat
283. Goat
284. Goat
285. Goat
286. Goat
287. Car
288. Goat
289. Goat
290. Goat
291. Goat
292. Goat
293. Goat
294. Car
295. Goat
296. Goat
297. Car
298. Car
299. Goat
300. Goat
301. Car
302. Goat
303. Car
304. Car
305. Goat
306. Car
307. Goat
308. Goat
309. Car
310. Goat
311. Goat
312. Goat
313. Goat
314. Goat
315. Car
316. Goat
317. Car
318. Goat
319. Car
320. Goat
321. Goat
322. Car
323. Goat
324. Car
325. Car
326. Goat
327. Car
328. Goat
329. Goat
330. Goat
331. Goat
332. Car
333. Goat
334. Car
335. Goat
336. Goat
337. Car
338. Car
339. Car
340. Car
341. Goat
342. Car
343. Car
344. Goat
345. Car
346. Car
347. Goat
348. Car
349. Goat
350. Car
351. Car
352. Goat
353. Goat
354. Goat
355. Goat
356. Goat
357. Goat
358. Goat
359. Car
360. Goat
361. Goat
362. Goat
363. Goat
364. Goat
365. Car
366. Goat
367. Goat
368. Goat
369. Goat
370. Car
371. Goat
372. Car
373. Goat
374. Car
375. Goat
376. Car
377. Goat
378. Car
379. Car
380. Goat
381. Goat
382. Goat
383. Goat
384. Car
385. Car
386. Goat
387. Goat
388. Goat
389. Car
390. Goat
391. Goat
392. Goat
393. Goat
394. Goat
395. Goat
396. Goat
397. Goat
398. Goat
399. Car
400. Goat
401. Goat
402. Goat
403. Car
404. Goat
405. Goat
406. Car
407. Goat
408. Car
409. Goat
410. Goat
411. Car
412. Car
413. Goat
414. Goat
415. Goat
416. Goat
417. Goat
418. Goat
419. Goat
420. Car
421. Goat
422. Car
423. Goat
424. Goat
425. Goat
426. Goat
427. Goat
428. Goat
429. Goat
430. Car
431. Goat
432. Goat
433. Car
434. Car
435. Goat
436. Goat
437. Goat
438. Goat
439. Goat
440. Goat
441. Goat
442. Goat
443. Goat
444. Goat
445. Goat
446. Goat
447. Goat
448. Goat
449. Goat
450. Car
451. Goat
452. Goat
453. Car
454. Goat
455. Goat
456. Goat
457. Goat
458. Goat
459. Car
460. Car
461. Car
462. Goat
463. Car
464. Goat
465. Goat
466. Goat
467. Goat
468. Goat
469. Goat
470. Goat
471. Goat
472. Goat
473. Goat
474. Goat
475. Goat
476. Goat
477. Car
478. Car
479. Goat
480. Goat
481. Goat
482. Goat
483. Goat
484. Goat
485. Car
486. Car
487. Goat
488. Car
489. Goat
490. Car
491. Car
492. Goat
493. Car
494. Goat
495. Goat
496. Goat
497. Goat
498. Goat
499. Car
500. Goat
501. Goat
502. Goat
503. Car
504. Goat
505. Goat
506. Goat
507. Goat
508. Goat
509. Car
510. Goat
511. Goat
512. Goat
513. Goat
514. Goat
515. Goat
516. Goat
517. Goat
518. Goat
519. Goat
520. Car
521. Car
522. Goat
523. Goat
524. Car
525. Goat
526. Goat
527. Goat
528. Goat
529. Goat
530. Goat
531. Car
532. Car
533. Car
534. Goat
535. Car
536. Goat
537. Car
538. Goat
539. Goat
540. Car
541. Goat
542. Goat
543. Car
544. Goat
545. Car
546. Goat
547. Goat
548. Goat
549. Goat
550. Car
551. Goat
552. Goat
553. Car
554. Goat
555. Goat
556. Goat
557. Goat
558. Goat
559. Goat
560. Goat
561. Car
562. Goat
563. Car
564. Goat
565. Goat
566. Goat
567. Goat
568. Goat
569. Car
570. Goat
571. Goat
572. Goat
573. Goat
574. Goat
575. Car
576. Car
577. Goat
578. Goat
579. Car
580. Goat
581. Car
582. Car
583. Goat
584. Car
585. Car
586. Goat
587. Goat
588. Goat
589. Goat
590. Car
591. Goat
592. Goat
593. Car
594. Goat
595. Car
596. Goat
597. Car
598. Goat
599. Goat
600. Goat
601. Goat
602. Goat
603. Car
604. Goat
605. Goat
606. Car
607. Goat
608. Goat
609. Goat
610. Goat
611. Goat
612. Car
613. Goat
614. Car
615. Goat
616. Goat
617. Car
618. Goat
619. Goat
620. Goat
621. Goat
622. Goat
623. Goat
624. Car
625. Car
626. Goat
627. Car
628. Goat
629. Goat
630. Goat
631. Goat
632. Car
633. Goat
634. Goat
635. Goat
636. Goat
637. Car
638. Car
639. Goat
640. Goat
641. Goat
642. Goat
643. Goat
644. Car
645. Goat
646. Goat
647. Goat
648. Goat
649. Goat
650. Goat
651. Car
652. Car
653. Goat
654. Goat
655. Goat
656. Car
657. Goat
658. Goat
659. Goat
660. Goat
661. Goat
662. Goat
663. Goat
664. Goat
665. Goat
666. Goat
667. Car
668. Goat
669. Goat
670. Goat
671. Car
672. Goat
673. Car
674. Goat
675. Goat
676. Goat
677. Goat
678. Goat
679. Goat
680. Car
681. Car
682. Car
683. Car
684. Goat
685. Goat
686. Car
687. Car
688. Goat
689. Car
690. Goat
691. Goat
692. Goat
693. Goat
694. Goat
695. Goat
696. Goat
697. Goat
698. Car
699. Car
700. Goat
701. Car
702. Goat
703. Car
704. Goat
705. Car
706. Car
707. Goat
708. Goat
709. Goat
710. Car
711. Car
712. Car
713. Car
714. Goat
715. Goat
716. Car
717. Goat
718. Goat
719. Goat
720. Car
721. Car
722. Goat
723. Goat
724. Goat
725. Goat
726. Goat
727. Goat
728. Goat
729. Goat
730. Goat
731. Car
732. Goat
733. Goat
734. Car
735. Goat
736. Goat
737. Goat
738. Goat
739. Goat
740. Goat
741. Car
742. Goat
743. Goat
744. Goat
745. Goat
746. Goat
747. Goat
748. Goat
749. Goat
750. Goat
751. Car
752. Car
753. Goat
754. Goat
755. Car
756. Car
757. Car
758. Goat
759. Car
760. Goat
761. Goat
762. Car
763. Goat
764. Car
765. Car
766. Goat
767. Car
768. Goat
769. Goat
770. Goat
771. Goat
772. Goat
773. Goat
774. Goat
775. Car
776. Goat
777. Goat
778. Car
779. Goat
780. Car
781. Goat
782. Goat
783. Car
784. Goat
785. Car
786. Goat
787. Car
788. Goat
789. Goat
790. Goat
791. Goat
792. Goat
793. Goat
794. Car
795. Goat
796. Car
797. Goat
798. Car
799. Goat
800. Goat
801. Goat
802. Goat
803. Goat
804. Goat
805. Goat
806. Car
807. Goat
808. Car
809. Goat
810. Car
811. Car
812. Goat
813. Goat
814. Goat
815. Goat
816. Car
817. Goat
818. Goat
819. Goat
820. Goat
821. Goat
822. Car
823. Car
824. Goat
825. Car
826. Car
827. Goat
828. Goat
829. Car
830. Goat
831. Goat
832. Car
833. Car
834. Goat
835. Car
836. Goat
837. Car
838. Goat
839. Car
840. Goat
841. Goat
842. Goat
843. Car
844. Goat
845. Car
846. Goat
847. Goat
848. Goat
849. Goat
850. Goat
851. Goat
852. Car
853. Goat
854. Goat
855. Goat
856. Car
857. Goat
858. Goat
859. Goat
860. Goat
861. Goat
862. Goat
863. Goat
864. Goat
865. Car
866. Car
867. Goat
868. Goat
869. Goat
870. Goat
871. Goat
872. Goat
873. Car
874. Car
875. Goat
876. Goat
877. Car
878. Goat
879. Goat
880. Goat
881. Car
882. Goat
883. Goat
884. Car
885. Car
886. Goat
887. Goat
888. Goat
889. Goat
890. Goat
891. Car
892. Car
893. Car
894. Car
895. Goat
896. Goat
897. Goat
898. Car
899. Car
900. Car
901. Car
902. Goat
903. Car
904. Car
905. Goat
906. Car
907. Goat
908. Car
909. Goat
910. Car
911. Car
912. Goat
913. Goat
914. Goat
915. Goat
916. Goat
917. Goat
918. Goat
919. Goat
920. Car
921. Car
922. Car
923. Goat
924. Goat
925. Goat
926. Goat
927. Goat
928. Goat
929. Goat
930. Goat
931. Goat
932. Goat
933. Goat
934. Car
935. Goat
936. Car
937. Goat
938. Goat
939. Car
940. Goat
941. Goat
942. Goat
943. Goat
944. Goat
945. Goat
946. Car
947. Car
948. Goat
949. Goat
950. Goat
951. Goat
952. Car
953. Goat
954. Car
955. Car
956. Goat
957. Goat
958. Goat
959. Car
960. Goat
961. Goat
962. Goat
963. Car
964. Goat
965. Goat
966. Goat
967. Car
968. Car
969. Goat
970. Goat
971. Car
972. Goat
973. Car
974. Goat
975. Goat
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Goat
982. Goat
983. Goat
984. Car
985. Car
986. Goat
987. Goat
988. Car
989. Goat
990. Goat
991. Car
992. Car
993. Goat
994. Goat
995. Goat
996. Car
997. Goat
998. Goat
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 89 times and lost 911 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 36 times and lost 64 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}