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. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Goat
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 336 times and lost 664 times.

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