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. Car
4. Goat
5. Goat
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 367 times and lost 633 times.

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