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. Car
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 340 times and lost 660 times.

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