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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
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 309 times and lost 691 times.

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