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. Car
6. Car
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 339 times and lost 661 times.

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