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. Goat
5. Car
6. Car
7. Goat
8. Car
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 370 times and lost 630 times.

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