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 2 times and lost 8 times.

1. Goat
2. Goat
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 352 times and lost 648 times.

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