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 2 times and lost 8 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Goat
6. Car
7. Goat
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 345 times and lost 655 times.

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