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. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Car
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 334 times and lost 666 times.

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