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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Goat
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 338 times and lost 662 times.

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