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 2 times and lost 8 times.

1. Goat
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 315 times and lost 685 times.

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