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. Car
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
7. Car
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 324 times and lost 676 times.

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