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

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