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 4 times and lost 6 times.

1. Goat
2. Car
3. Goat
4. Car
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 322 times and lost 678 times.

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