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. Goat
3. Car
4. Goat
5. Goat
6. Car
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 337 times and lost 663 times.

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