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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
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 340 times and lost 660 times.

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