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 3 times and lost 7 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Car
9. Goat
10. Car

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 339 times and lost 661 times.

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