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 3 times and lost 7 times.

1. Car
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Car
9. Goat
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 310 times and lost 690 times.

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