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. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Car
7. Car
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 343 times and lost 657 times.

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