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 4 times and lost 6 times.

1. Car
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Car
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 340 times and lost 660 times.

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