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. Car
3. Goat
4. Car
5. Car
6. Goat
7. Car
8. Goat
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 356 times and lost 644 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Car
6. Car
7. Goat
8. Goat
9. Goat
10. Goat
11. Goat
12. Goat
13. Goat
14. Car
15. Goat
16. Car
17. Car
18. Goat
19. Goat
20. Goat
21. Car
22. Car
23. Goat
24. Car
25. Car
26. Goat
27. Car
28. Goat
29. Goat
30. Car
31. Goat
32. Goat
33. Goat
34. Goat
35. Goat
36. Goat
37. Car
38. Car
39. Goat
40. Goat
41. Car
42. Goat
43. Goat
44. Goat
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. Car
58. Goat
59. Goat
60. Car
61. Car
62. Goat
63. Goat
64. Car
65. Goat
66. Car
67. Goat
68. Goat
69. Goat
70. Car
71. Goat
72. Goat
73. Goat
74. Goat
75. Car
76. Goat
77. Car
78. Goat
79. Goat
80. Goat
81. Goat
82. Car
83. Goat
84. Goat
85. Goat
86. Car
87. Car
88. Goat
89. Goat
90. Goat
91. Car
92. Goat
93. Car
94. Car
95. Car
96. Goat
97. Goat
98. Car
99. Goat
100. Goat
101. Car
102. Goat
103. Goat
104. Goat
105. Car
106. Goat
107. Goat
108. Car
109. Goat
110. Goat
111. Goat
112. Goat
113. Goat
114. Goat
115. Car
116. Goat
117. Goat
118. Goat
119. Goat
120. Goat
121. Goat
122. Goat
123. Car
124. Goat
125. Goat
126. Goat
127. Goat
128. Goat
129. Car
130. Goat
131. Goat
132. Car
133. Car
134. Goat
135. Goat
136. Goat
137. Goat
138. Car
139. Car
140. Car
141. Car
142. Goat
143. Goat
144. Car
145. Goat
146. Car
147. Goat
148. Goat
149. Goat
150. Goat
151. Goat
152. Goat
153. Goat
154. Goat
155. Goat
156. Car
157. Car
158. Goat
159. Car
160. Goat
161. Goat
162. Goat
163. Goat
164. Car
165. Car
166. Goat
167. Car
168. Goat
169. Goat
170. Car
171. Goat
172. Car
173. Car
174. Goat
175. Car
176. Goat
177. Goat
178. Goat
179. Goat
180. Goat
181. Car
182. Car
183. Car
184. Car
185. Goat
186. Goat
187. Goat
188. Goat
189. Goat
190. Goat
191. Goat
192. Car
193. Car
194. Goat
195. Goat
196. Goat
197. Car
198. Goat
199. Goat
200. Goat
201. Car
202. Goat
203. Goat
204. Car
205. Goat
206. Goat
207. Goat
208. Car
209. Goat
210. Goat
211. Goat
212. Goat
213. Car
214. Goat
215. Goat
216. Goat
217. Car
218. Car
219. Goat
220. Goat
221. Goat
222. Car
223. Car
224. Goat
225. Goat
226. Goat
227. Goat
228. Goat
229. Car
230. Goat
231. Goat
232. Car
233. Goat
234. Car
235. Car
236. Car
237. Goat
238. Car
239. Goat
240. Goat
241. Goat
242. Car
243. Goat
244. Goat
245. Car
246. Goat
247. Car
248. Goat
249. Goat
250. Car
251. Goat
252. Goat
253. Goat
254. Goat
255. Goat
256. Goat
257. Goat
258. Car
259. Goat
260. Car
261. Goat
262. Goat
263. Goat
264. Car
265. Car
266. Goat
267. Goat
268. Goat
269. Goat
270. Goat
271. Car
272. Goat
273. Car
274. Car
275. Goat
276. Goat
277. Car
278. Goat
279. Goat
280. Goat
281. Goat
282. Goat
283. Goat
284. Car
285. Goat
286. Goat
287. Car
288. Car
289. Goat
290. Car
291. Goat
292. Goat
293. Goat
294. Goat
295. Goat
296. Car
297. Goat
298. Goat
299. Car
300. Goat
301. Car
302. Car
303. Car
304. Goat
305. Goat
306. Car
307. Car
308. Car
309. Car
310. Car
311. Goat
312. Car
313. Car
314. Goat
315. Goat
316. Goat
317. Car
318. Goat
319. Car
320. Car
321. Goat
322. Goat
323. Car
324. Goat
325. Goat
326. Car
327. Goat
328. Car
329. Goat
330. Goat
331. Goat
332. Goat
333. Car
334. Goat
335. Goat
336. Goat
337. Car
338. Car
339. Car
340. Goat
341. Car
342. Goat
343. Car
344. Goat
345. Car
346. Car
347. Goat
348. Car
349. Car
350. Goat
351. Goat
352. Car
353. Goat
354. Car
355. Goat
356. Goat
357. Goat
358. Car
359. Car
360. Goat
361. Car
362. Car
363. Goat
364. Goat
365. Goat
366. Goat
367. Goat
368. Goat
369. Goat
370. Goat
371. Car
372. Goat
373. Car
374. Goat
375. Goat
376. Car
377. Goat
378. Goat
379. Goat
380. Goat
381. Car
382. Car
383. Goat
384. Goat
385. Car
386. Goat
387. Goat
388. Car
389. Goat
390. Car
391. Goat
392. Goat
393. Car
394. Car
395. Goat
396. Goat
397. Goat
398. Car
399. Goat
400. Car
401. Goat
402. Car
403. Goat
404. Goat
405. Goat
406. Goat
407. Car
408. Car
409. Goat
410. Goat
411. Car
412. Car
413. Goat
414. Car
415. Goat
416. Car
417. Goat
418. Car
419. Car
420. Car
421. Goat
422. Goat
423. Car
424. Car
425. Goat
426. Car
427. Car
428. Goat
429. Goat
430. Goat
431. Car
432. Goat
433. Goat
434. Car
435. Goat
436. Car
437. Goat
438. Goat
439. Goat
440. Goat
441. Goat
442. Goat
443. Goat
444. Goat
445. Car
446. Goat
447. Car
448. Goat
449. Car
450. Car
451. Goat
452. Car
453. Goat
454. Car
455. Car
456. Car
457. Car
458. Car
459. Goat
460. Car
461. Goat
462. Goat
463. Goat
464. Goat
465. Goat
466. Goat
467. Car
468. Goat
469. Goat
470. Goat
471. Goat
472. Goat
473. Car
474. Goat
475. Car
476. Goat
477. Goat
478. Goat
479. Goat
480. Goat
481. Goat
482. Goat
483. Goat
484. Goat
485. Goat
486. Goat
487. Goat
488. Car
489. Car
490. Goat
491. Goat
492. Goat
493. Goat
494. Car
495. Car
496. Goat
497. Goat
498. Goat
499. Goat
500. Goat
501. Car
502. Car
503. Car
504. Goat
505. Goat
506. Car
507. Goat
508. Goat
509. Goat
510. Goat
511. Goat
512. Car
513. Goat
514. Car
515. Goat
516. Goat
517. Goat
518. Car
519. Goat
520. Goat
521. Car
522. Goat
523. Goat
524. Car
525. Car
526. Car
527. Goat
528. Car
529. Goat
530. Goat
531. Goat
532. Goat
533. Car
534. Car
535. Goat
536. Goat
537. Goat
538. Goat
539. Goat
540. Goat
541. Goat
542. Goat
543. Goat
544. Car
545. Car
546. Car
547. Goat
548. Car
549. Goat
550. Car
551. Car
552. Goat
553. Goat
554. Goat
555. Goat
556. Goat
557. Goat
558. Car
559. Goat
560. Car
561. Goat
562. Goat
563. Goat
564. Goat
565. Goat
566. Car
567. Car
568. Goat
569. Car
570. Goat
571. Goat
572. Car
573. Car
574. Goat
575. Goat
576. Goat
577. Car
578. Goat
579. Car
580. Goat
581. Car
582. Goat
583. Car
584. Goat
585. Car
586. Car
587. Goat
588. Goat
589. Car
590. Car
591. Goat
592. Goat
593. Car
594. Car
595. Goat
596. Goat
597. Goat
598. Goat
599. Goat
600. Goat
601. Car
602. Car
603. Car
604. Car
605. Goat
606. Goat
607. Goat
608. Car
609. Car
610. Goat
611. Goat
612. Car
613. Car
614. Car
615. Goat
616. Car
617. Goat
618. Goat
619. Car
620. Car
621. Goat
622. Goat
623. Goat
624. Goat
625. Goat
626. Goat
627. Goat
628. Goat
629. Goat
630. Car
631. Goat
632. Goat
633. Car
634. Goat
635. Goat
636. Goat
637. Goat
638. Car
639. Goat
640. Car
641. Goat
642. Goat
643. Car
644. Goat
645. Goat
646. Goat
647. Goat
648. Goat
649. Car
650. Goat
651. Goat
652. Goat
653. Goat
654. Car
655. Goat
656. Goat
657. Car
658. Car
659. Car
660. Goat
661. Goat
662. Goat
663. Car
664. Goat
665. Goat
666. Goat
667. Goat
668. Goat
669. Goat
670. Car
671. Car
672. Car
673. Goat
674. Car
675. Goat
676. Goat
677. Car
678. Goat
679. Goat
680. Car
681. Goat
682. Goat
683. Car
684. Goat
685. Car
686. Car
687. Goat
688. Goat
689. Goat
690. Goat
691. Car
692. Car
693. Goat
694. Goat
695. Goat
696. Goat
697. Car
698. Goat
699. Car
700. Goat
701. Goat
702. Goat
703. Car
704. Car
705. Car
706. Car
707. Car
708. Car
709. Car
710. Goat
711. Car
712. Goat
713. Goat
714. Goat
715. Car
716. Goat
717. Goat
718. Goat
719. Goat
720. Goat
721. Car
722. Car
723. Goat
724. Car
725. Car
726. Goat
727. Car
728. Goat
729. Goat
730. Goat
731. Goat
732. Goat
733. Car
734. Goat
735. Goat
736. Car
737. Goat
738. Car
739. Goat
740. Goat
741. Car
742. Goat
743. Car
744. Car
745. Car
746. Goat
747. Goat
748. Goat
749. Car
750. Goat
751. Goat
752. Goat
753. Car
754. Car
755. Car
756. Goat
757. Goat
758. Goat
759. Car
760. Car
761. Goat
762. Car
763. Goat
764. Goat
765. Car
766. Car
767. Goat
768. Car
769. Car
770. Goat
771. Goat
772. Goat
773. Car
774. Goat
775. Car
776. Goat
777. Goat
778. Goat
779. Car
780. Goat
781. Car
782. Goat
783. Goat
784. Goat
785. Car
786. Car
787. Goat
788. Car
789. Car
790. Goat
791. Goat
792. Goat
793. Goat
794. Goat
795. Goat
796. Car
797. Car
798. Goat
799. Goat
800. Goat
801. Goat
802. Goat
803. Goat
804. Goat
805. Goat
806. Goat
807. Goat
808. Car
809. Car
810. Goat
811. Goat
812. Car
813. Goat
814. Goat
815. Goat
816. Goat
817. Goat
818. Goat
819. Goat
820. Goat
821. Goat
822. Car
823. Goat
824. Car
825. Goat
826. Goat
827. Car
828. Goat
829. Goat
830. Car
831. Car
832. Car
833. Car
834. Goat
835. Car
836. Car
837. Goat
838. Car
839. Goat
840. Goat
841. Car
842. Goat
843. Goat
844. Car
845. Goat
846. Car
847. Goat
848. Goat
849. Goat
850. Goat
851. Car
852. Goat
853. Car
854. Car
855. Goat
856. Goat
857. Goat
858. Goat
859. Goat
860. Car
861. Goat
862. Car
863. Car
864. Goat
865. Car
866. Goat
867. Goat
868. Goat
869. Goat
870. Car
871. Goat
872. Goat
873. Car
874. Goat
875. Car
876. Goat
877. Goat
878. Goat
879. Goat
880. Car
881. Goat
882. Goat
883. Goat
884. Goat
885. Goat
886. Car
887. Goat
888. Goat
889. Car
890. Goat
891. Goat
892. Car
893. Car
894. Goat
895. Car
896. Goat
897. Goat
898. Car
899. Goat
900. Car
901. Goat
902. Goat
903. Goat
904. Goat
905. Goat
906. Goat
907. Goat
908. Car
909. Goat
910. Goat
911. Goat
912. Goat
913. Goat
914. Goat
915. Car
916. Goat
917. Goat
918. Goat
919. Goat
920. Goat
921. Car
922. Goat
923. Car
924. Car
925. Goat
926. Goat
927. Car
928. Goat
929. Goat
930. Goat
931. Car
932. Goat
933. Car
934. Goat
935. Car
936. Car
937. Goat
938. Goat
939. Goat
940. Goat
941. Goat
942. Car
943. Goat
944. Goat
945. Car
946. Goat
947. Car
948. Goat
949. Car
950. Goat
951. Goat
952. Car
953. Car
954. Goat
955. Car
956. Goat
957. Goat
958. Goat
959. Goat
960. Car
961. Goat
962. Goat
963. Car
964. Goat
965. Goat
966. Goat
967. Car
968. Car
969. Goat
970. Car
971. Car
972. Goat
973. Goat
974. Goat
975. Goat
976. Car
977. Car
978. Goat
979. Goat
980. Goat
981. Goat
982. Goat
983. Goat
984. Car
985. Goat
986. Goat
987. Goat
988. Goat
989. Car
990. Car
991. Goat
992. Goat
993. Goat
994. Goat
995. Goat
996. Goat
997. Car
998. Car
999. Goat
1000. Car

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 127 times and lost 873 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 33 times and lost 67 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;
	}
}