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. Car
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
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 343 times and lost 657 times.

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