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. Car
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
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 351 times and lost 649 times.

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