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 6 times and lost 4 times.

1. Goat
2. Car
3. Car
4. Car
5. Goat
6. Car
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 311 times and lost 689 times.

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