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