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. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Car

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 350 times and lost 650 times.

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