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. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Car
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 332 times and lost 668 times.

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