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. Car
4. Goat
5. Car
6. Goat
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 327 times and lost 673 times.

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