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 4 times and lost 6 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Car
9. Car
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 317 times and lost 683 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
9. Car
10. Goat
11. Car
12. Car
13. Goat
14. Car
15. Goat
16. Goat
17. Goat
18. Goat
19. Goat
20. Goat
21. Goat
22. Car
23. Car
24. Goat
25. Goat
26. Car
27. Goat
28. Goat
29. Car
30. Car
31. Car
32. Goat
33. Goat
34. Car
35. Goat
36. Car
37. Goat
38. Car
39. Goat
40. Car
41. Goat
42. Car
43. Car
44. Goat
45. Goat
46. Goat
47. Car
48. Car
49. Goat
50. Goat
51. Goat
52. Goat
53. Goat
54. Goat
55. Goat
56. Goat
57. Car
58. Goat
59. Car
60. Goat
61. Goat
62. Goat
63. Car
64. Goat
65. Goat
66. Goat
67. Goat
68. Goat
69. Goat
70. Goat
71. Goat
72. Goat
73. Car
74. Goat
75. Car
76. Goat
77. Goat
78. Car
79. Car
80. Goat
81. Goat
82. Goat
83. Goat
84. Car
85. Goat
86. Goat
87. Car
88. Goat
89. Goat
90. Car
91. Car
92. Goat
93. Goat
94. Goat
95. Goat
96. Goat
97. Car
98. Car
99. Goat
100. Car
101. Goat
102. Car
103. Goat
104. Goat
105. Car
106. Car
107. Goat
108. Goat
109. Goat
110. Goat
111. Goat
112. Goat
113. Goat
114. Goat
115. Goat
116. Car
117. Goat
118. Car
119. Goat
120. Goat
121. Goat
122. Goat
123. Goat
124. Car
125. Goat
126. Car
127. Goat
128. Goat
129. Goat
130. Goat
131. Goat
132. Goat
133. Goat
134. Goat
135. Car
136. Goat
137. Car
138. Goat
139. Car
140. Goat
141. Goat
142. Goat
143. Goat
144. Goat
145. Car
146. Goat
147. Goat
148. Goat
149. Goat
150. Goat
151. Goat
152. Goat
153. Goat
154. Car
155. Car
156. Goat
157. Goat
158. Goat
159. Goat
160. Goat
161. Goat
162. Goat
163. Goat
164. Goat
165. Car
166. Car
167. Car
168. Car
169. Goat
170. Car
171. Goat
172. Goat
173. Goat
174. Car
175. Goat
176. Goat
177. Goat
178. Goat
179. Goat
180. Goat
181. Goat
182. Car
183. Goat
184. Car
185. Goat
186. Car
187. Car
188. Goat
189. Car
190. Goat
191. Goat
192. Car
193. Goat
194. Goat
195. Goat
196. Goat
197. Car
198. Goat
199. Goat
200. Goat
201. Goat
202. Goat
203. Car
204. Goat
205. Goat
206. Goat
207. Car
208. Goat
209. Car
210. Car
211. Car
212. Car
213. Goat
214. Goat
215. Goat
216. Car
217. Goat
218. Car
219. Car
220. Goat
221. Car
222. Goat
223. Car
224. Goat
225. Goat
226. Car
227. Goat
228. Car
229. Goat
230. Car
231. Goat
232. Goat
233. Car
234. Goat
235. Car
236. Goat
237. Car
238. Goat
239. Car
240. Goat
241. Goat
242. Goat
243. Goat
244. Car
245. Goat
246. Goat
247. Goat
248. Goat
249. Goat
250. Goat
251. Goat
252. Goat
253. Goat
254. Goat
255. Car
256. Goat
257. Car
258. Goat
259. Goat
260. Car
261. Goat
262. Goat
263. Goat
264. Car
265. Car
266. Goat
267. Car
268. Goat
269. Car
270. Goat
271. Goat
272. Goat
273. Car
274. Goat
275. Goat
276. Car
277. Car
278. Goat
279. Goat
280. Goat
281. Car
282. Goat
283. Goat
284. Car
285. Goat
286. Goat
287. Goat
288. Car
289. Goat
290. Goat
291. Goat
292. Goat
293. Car
294. Car
295. Goat
296. Goat
297. Goat
298. Goat
299. Goat
300. Goat
301. Car
302. Goat
303. Car
304. Car
305. Goat
306. Goat
307. Goat
308. Goat
309. Goat
310. Car
311. Goat
312. Goat
313. Goat
314. Goat
315. Goat
316. Car
317. Goat
318. Goat
319. Goat
320. Car
321. Car
322. Goat
323. Goat
324. Goat
325. Goat
326. Goat
327. Goat
328. Car
329. Goat
330. Goat
331. Car
332. Goat
333. Goat
334. Goat
335. Car
336. Goat
337. Goat
338. Goat
339. Goat
340. Car
341. Goat
342. Car
343. Goat
344. Goat
345. Car
346. Car
347. Goat
348. Goat
349. Goat
350. Car
351. Car
352. Goat
353. Goat
354. Car
355. Goat
356. Goat
357. Car
358. Goat
359. Goat
360. Goat
361. Goat
362. Car
363. Goat
364. Goat
365. Goat
366. Goat
367. Goat
368. Goat
369. Car
370. Goat
371. Goat
372. Car
373. Car
374. Car
375. Goat
376. Car
377. Goat
378. Goat
379. Car
380. Car
381. Car
382. Car
383. Car
384. Car
385. Goat
386. Car
387. Car
388. Goat
389. Goat
390. Goat
391. Goat
392. Car
393. Car
394. Goat
395. Car
396. Car
397. Goat
398. Car
399. Car
400. Goat
401. Goat
402. Car
403. Goat
404. Goat
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. Goat
418. Goat
419. Goat
420. Goat
421. Goat
422. Goat
423. Goat
424. Goat
425. Car
426. Goat
427. Goat
428. Car
429. Goat
430. Goat
431. Goat
432. Goat
433. Goat
434. Goat
435. Goat
436. Car
437. Goat
438. Goat
439. Car
440. Goat
441. Goat
442. Goat
443. Car
444. Goat
445. Goat
446. Goat
447. Goat
448. Car
449. Car
450. Car
451. Car
452. Goat
453. Goat
454. Goat
455. Car
456. Car
457. Goat
458. Car
459. Goat
460. Goat
461. Goat
462. Goat
463. Car
464. Goat
465. Goat
466. Goat
467. Car
468. Goat
469. Goat
470. Car
471. Goat
472. Car
473. Goat
474. Goat
475. Goat
476. Car
477. Car
478. Goat
479. Goat
480. Car
481. Car
482. Goat
483. Car
484. Car
485. Goat
486. Goat
487. Goat
488. Goat
489. Goat
490. Car
491. Goat
492. Goat
493. Car
494. Goat
495. Goat
496. Goat
497. Goat
498. Goat
499. Goat
500. Goat
501. Goat
502. Goat
503. Goat
504. Car
505. Goat
506. Car
507. Goat
508. Goat
509. Goat
510. Goat
511. Goat
512. Goat
513. Goat
514. Goat
515. Goat
516. Car
517. Goat
518. Goat
519. Car
520. Car
521. Goat
522. Goat
523. Car
524. Goat
525. Goat
526. Goat
527. Goat
528. Goat
529. Goat
530. Goat
531. Car
532. Car
533. Goat
534. Goat
535. Car
536. Car
537. Goat
538. Goat
539. Goat
540. Goat
541. Goat
542. Goat
543. Goat
544. Goat
545. Car
546. Goat
547. Goat
548. Car
549. Goat
550. Goat
551. Goat
552. Goat
553. Goat
554. Goat
555. Goat
556. Car
557. Car
558. Goat
559. Car
560. Goat
561. Goat
562. Goat
563. Goat
564. Goat
565. Car
566. Goat
567. Goat
568. Car
569. Goat
570. Car
571. Goat
572. Goat
573. Goat
574. Goat
575. Goat
576. Car
577. Car
578. Goat
579. Goat
580. Goat
581. Car
582. Goat
583. Goat
584. Goat
585. Car
586. Goat
587. Goat
588. Goat
589. Car
590. Goat
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. Goat
605. Goat
606. Goat
607. Car
608. Goat
609. Car
610. Goat
611. Goat
612. Car
613. Goat
614. Goat
615. Goat
616. Goat
617. Car
618. Goat
619. Goat
620. Goat
621. Goat
622. Goat
623. Goat
624. Goat
625. Car
626. Car
627. Goat
628. Car
629. Car
630. Goat
631. Car
632. Goat
633. Goat
634. Goat
635. Goat
636. Car
637. Car
638. Goat
639. Goat
640. Goat
641. Goat
642. Goat
643. Car
644. Car
645. Goat
646. Goat
647. Car
648. Car
649. Goat
650. Goat
651. Goat
652. Goat
653. Goat
654. Goat
655. Goat
656. Car
657. Goat
658. Car
659. Goat
660. Car
661. Goat
662. Goat
663. Goat
664. Goat
665. Goat
666. Goat
667. Goat
668. Goat
669. Goat
670. Car
671. Goat
672. Car
673. Goat
674. Car
675. Goat
676. Goat
677. Goat
678. Car
679. Goat
680. Goat
681. Car
682. Goat
683. Car
684. Car
685. Goat
686. Car
687. Goat
688. Goat
689. Car
690. Goat
691. Car
692. Goat
693. Car
694. Car
695. Goat
696. Goat
697. Goat
698. Car
699. Goat
700. Car
701. Goat
702. Goat
703. Car
704. Goat
705. Goat
706. Goat
707. Car
708. Goat
709. Goat
710. Goat
711. Car
712. Car
713. Goat
714. Car
715. Car
716. Goat
717. Goat
718. Goat
719. Goat
720. Car
721. Goat
722. Goat
723. Goat
724. Goat
725. Car
726. Car
727. Goat
728. Goat
729. Car
730. Goat
731. Goat
732. Goat
733. Goat
734. Goat
735. Goat
736. Car
737. Car
738. Goat
739. Car
740. Goat
741. Car
742. Goat
743. Goat
744. Goat
745. Car
746. Car
747. Goat
748. Goat
749. Car
750. Goat
751. Car
752. Car
753. Car
754. Car
755. Car
756. Goat
757. Goat
758. Goat
759. Goat
760. Goat
761. Goat
762. Goat
763. Car
764. Goat
765. Goat
766. Car
767. Goat
768. Goat
769. Car
770. Goat
771. Goat
772. Goat
773. Car
774. Car
775. Goat
776. Goat
777. Car
778. Goat
779. Car
780. Goat
781. Car
782. Goat
783. Goat
784. Car
785. Goat
786. Goat
787. Goat
788. Goat
789. Car
790. Goat
791. Goat
792. Car
793. Goat
794. Car
795. Goat
796. Goat
797. Goat
798. Goat
799. Car
800. Goat
801. Car
802. Goat
803. Goat
804. Car
805. Goat
806. Car
807. Goat
808. Goat
809. Goat
810. Goat
811. Goat
812. Goat
813. Goat
814. Goat
815. Goat
816. Goat
817. Goat
818. Car
819. Car
820. Goat
821. Goat
822. Car
823. Goat
824. Car
825. Goat
826. Goat
827. Goat
828. Goat
829. Goat
830. Goat
831. Goat
832. Goat
833. Goat
834. Car
835. Goat
836. Goat
837. Car
838. Goat
839. Goat
840. Goat
841. Car
842. Goat
843. Car
844. Goat
845. Car
846. Car
847. Goat
848. Goat
849. Car
850. Goat
851. Goat
852. Car
853. Goat
854. Goat
855. Goat
856. Car
857. Car
858. Goat
859. Car
860. Goat
861. Car
862. Goat
863. Goat
864. Goat
865. Goat
866. Goat
867. Car
868. Goat
869. Goat
870. Goat
871. Goat
872. Goat
873. Goat
874. Goat
875. Goat
876. Goat
877. Car
878. Goat
879. Goat
880. Car
881. Car
882. Goat
883. Goat
884. Goat
885. Goat
886. Goat
887. Car
888. Car
889. Goat
890. Goat
891. Goat
892. Car
893. Car
894. Goat
895. Car
896. Goat
897. Car
898. Goat
899. Goat
900. Car
901. Goat
902. Goat
903. Goat
904. Goat
905. Car
906. Goat
907. Goat
908. Car
909. Car
910. Goat
911. Goat
912. Goat
913. Car
914. Car
915. Goat
916. Car
917. Goat
918. Goat
919. Goat
920. Goat
921. Goat
922. Goat
923. Car
924. Car
925. Goat
926. Car
927. Goat
928. Car
929. Goat
930. Car
931. Goat
932. Goat
933. Car
934. Car
935. Car
936. Car
937. Goat
938. Car
939. Goat
940. Car
941. Goat
942. Goat
943. Car
944. Goat
945. Goat
946. Goat
947. Goat
948. Goat
949. Goat
950. Goat
951. Car
952. Car
953. Goat
954. Goat
955. Goat
956. Car
957. Goat
958. Car
959. Goat
960. Goat
961. Goat
962. Goat
963. Goat
964. Goat
965. Car
966. Goat
967. Goat
968. Goat
969. Car
970. Goat
971. Car
972. Car
973. Goat
974. Goat
975. Car
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Goat
982. Goat
983. Goat
984. Goat
985. Goat
986. Car
987. Goat
988. Goat
989. Car
990. Car
991. Goat
992. Car
993. Goat
994. Goat
995. Goat
996. Goat
997. Goat
998. Car
999. Car
1000. Car

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 97 times and lost 903 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 36 times and lost 64 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;
	}
}