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. Car
5. Goat
6. Goat
7. Goat
8. Goat
9. Car
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 366 times and lost 634 times.

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