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 5 times and lost 5 times.

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