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 3 times and lost 7 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Car
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 326 times and lost 674 times.

1. Goat
2. Car
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Car
10. Car
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. Car
28. Car
29. Goat
30. Goat
31. Goat
32. Goat
33. Goat
34. Car
35. Goat
36. Goat
37. Goat
38. Goat
39. Goat
40. Goat
41. Goat
42. Goat
43. Car
44. Goat
45. Goat
46. Goat
47. Goat
48. Goat
49. Goat
50. Goat
51. Goat
52. Car
53. Goat
54. Goat
55. Car
56. Goat
57. Goat
58. Goat
59. Goat
60. Car
61. Car
62. Goat
63. Car
64. Goat
65. Goat
66. Car
67. Goat
68. Goat
69. Goat
70. Goat
71. Goat
72. Goat
73. Car
74. Car
75. Goat
76. Car
77. Goat
78. Goat
79. Goat
80. Goat
81. Goat
82. Goat
83. Goat
84. Goat
85. Goat
86. Goat
87. Goat
88. Goat
89. Goat
90. Car
91. Goat
92. Car
93. Car
94. Car
95. Goat
96. Goat
97. Goat
98. Car
99. Goat
100. Car
101. Goat
102. Goat
103. Car
104. Goat
105. Goat
106. Goat
107. Goat
108. Car
109. Goat
110. Car
111. Car
112. Goat
113. Goat
114. Car
115. Goat
116. Goat
117. Goat
118. Car
119. Goat
120. Goat
121. Goat
122. Car
123. Goat
124. Car
125. Car
126. Goat
127. Goat
128. Car
129. Car
130. Car
131. Car
132. Car
133. Goat
134. Car
135. Goat
136. Car
137. Goat
138. Goat
139. Car
140. Goat
141. Goat
142. Car
143. Goat
144. Goat
145. Goat
146. Goat
147. Goat
148. Goat
149. Car
150. Car
151. Goat
152. Goat
153. Goat
154. Car
155. Goat
156. Car
157. Car
158. Goat
159. Goat
160. Goat
161. Goat
162. Car
163. Goat
164. Goat
165. Goat
166. Goat
167. Goat
168. Goat
169. Car
170. Goat
171. Goat
172. Goat
173. Goat
174. Goat
175. Goat
176. Goat
177. Goat
178. Goat
179. Car
180. Goat
181. Goat
182. Goat
183. Car
184. Goat
185. Car
186. Goat
187. Car
188. Car
189. Goat
190. Car
191. Goat
192. Goat
193. Car
194. Goat
195. Goat
196. Car
197. Car
198. Goat
199. Car
200. Goat
201. Goat
202. Goat
203. Car
204. Goat
205. Car
206. Car
207. Car
208. Goat
209. Car
210. Car
211. Goat
212. Goat
213. Goat
214. Goat
215. Car
216. Goat
217. Goat
218. Car
219. Goat
220. Car
221. Car
222. Goat
223. Car
224. Goat
225. Car
226. Goat
227. Goat
228. Goat
229. Goat
230. Goat
231. Goat
232. Car
233. Car
234. Goat
235. Goat
236. Goat
237. Goat
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Goat
244. Car
245. Car
246. Goat
247. Car
248. Goat
249. Goat
250. Car
251. Goat
252. Goat
253. Car
254. Goat
255. Car
256. Goat
257. Goat
258. Goat
259. Goat
260. Goat
261. Car
262. Goat
263. Car
264. Goat
265. Car
266. Goat
267. Goat
268. Goat
269. Goat
270. Car
271. Car
272. Car
273. Car
274. Goat
275. Goat
276. Goat
277. Goat
278. Goat
279. Goat
280. Goat
281. Goat
282. Goat
283. Goat
284. Goat
285. Car
286. Car
287. Goat
288. Car
289. Car
290. Car
291. Goat
292. Goat
293. Car
294. Goat
295. Goat
296. Car
297. Goat
298. Goat
299. Goat
300. Car
301. Goat
302. Car
303. Goat
304. Car
305. Goat
306. Goat
307. Car
308. Car
309. Goat
310. Goat
311. Goat
312. Car
313. Car
314. Goat
315. Goat
316. Goat
317. Car
318. Car
319. Car
320. Car
321. Goat
322. Car
323. Goat
324. Car
325. Goat
326. Car
327. Car
328. Car
329. Goat
330. Goat
331. Goat
332. Goat
333. Goat
334. Goat
335. Car
336. Goat
337. Car
338. Car
339. Goat
340. Car
341. Goat
342. Goat
343. Car
344. Goat
345. Car
346. Goat
347. Car
348. Goat
349. Car
350. Goat
351. Goat
352. Car
353. Goat
354. Goat
355. Goat
356. Goat
357. Goat
358. Car
359. Car
360. Goat
361. Goat
362. Car
363. Goat
364. Car
365. Goat
366. Car
367. Car
368. Car
369. Car
370. Goat
371. Goat
372. Goat
373. Goat
374. Car
375. Goat
376. Goat
377. Car
378. Goat
379. Goat
380. Goat
381. Car
382. Car
383. Car
384. Goat
385. Goat
386. Car
387. Goat
388. Goat
389. Car
390. Car
391. Goat
392. Goat
393. Goat
394. Goat
395. Goat
396. Goat
397. Car
398. Car
399. Goat
400. Goat
401. Goat
402. Car
403. Car
404. Goat
405. Goat
406. Goat
407. Goat
408. Car
409. Car
410. Goat
411. Goat
412. Car
413. Goat
414. Car
415. Goat
416. Car
417. Goat
418. Goat
419. Car
420. Goat
421. Goat
422. Goat
423. Goat
424. Goat
425. Goat
426. Goat
427. Car
428. Goat
429. Goat
430. Goat
431. Car
432. Goat
433. Goat
434. Car
435. Goat
436. Goat
437. Car
438. Goat
439. Car
440. Goat
441. Car
442. Car
443. Car
444. Goat
445. Goat
446. Goat
447. Goat
448. Goat
449. Goat
450. Goat
451. Goat
452. Car
453. Goat
454. Goat
455. Goat
456. Goat
457. Goat
458. Goat
459. Goat
460. Goat
461. Car
462. Car
463. Car
464. Goat
465. Goat
466. Car
467. Goat
468. Goat
469. Goat
470. Goat
471. Goat
472. Car
473. Car
474. Goat
475. Goat
476. Car
477. Goat
478. Car
479. Goat
480. Car
481. Goat
482. Car
483. Car
484. Goat
485. Goat
486. Goat
487. Goat
488. Goat
489. Car
490. Car
491. Goat
492. Goat
493. Car
494. Goat
495. Car
496. Goat
497. Goat
498. Car
499. Car
500. Goat
501. Goat
502. Car
503. Goat
504. Goat
505. Goat
506. Goat
507. Car
508. Goat
509. Goat
510. Car
511. Car
512. Goat
513. Goat
514. Goat
515. Car
516. Goat
517. Goat
518. Car
519. Goat
520. Goat
521. Goat
522. Goat
523. Goat
524. Goat
525. Goat
526. Goat
527. Goat
528. Car
529. Goat
530. Goat
531. Goat
532. Goat
533. Goat
534. Car
535. Car
536. Goat
537. Car
538. Car
539. Goat
540. Goat
541. Goat
542. Car
543. Goat
544. Goat
545. Goat
546. Goat
547. Car
548. Car
549. Goat
550. Goat
551. Car
552. Car
553. Car
554. Car
555. Goat
556. Car
557. Goat
558. Goat
559. Goat
560. Goat
561. Car
562. Goat
563. Goat
564. Car
565. Car
566. Goat
567. Car
568. Goat
569. Car
570. Car
571. Goat
572. Goat
573. Car
574. Car
575. Goat
576. Car
577. Car
578. Goat
579. Goat
580. Goat
581. Car
582. Goat
583. Goat
584. Goat
585. Goat
586. Goat
587. Goat
588. Goat
589. Goat
590. Goat
591. Goat
592. Car
593. Car
594. Goat
595. Car
596. Goat
597. Goat
598. Goat
599. Goat
600. Goat
601. Goat
602. Goat
603. Goat
604. Goat
605. Goat
606. Goat
607. Goat
608. Goat
609. Goat
610. Car
611. Goat
612. Car
613. Goat
614. Goat
615. Car
616. Goat
617. Goat
618. Goat
619. Goat
620. Goat
621. Goat
622. Car
623. Goat
624. Goat
625. Goat
626. Goat
627. Goat
628. Goat
629. Goat
630. Goat
631. Goat
632. Goat
633. Goat
634. Goat
635. Car
636. Goat
637. Goat
638. Car
639. Goat
640. Goat
641. Goat
642. Goat
643. Goat
644. Car
645. Goat
646. Goat
647. Car
648. Goat
649. Goat
650. Goat
651. Goat
652. Car
653. Goat
654. Goat
655. Car
656. Car
657. Car
658. Goat
659. Car
660. Car
661. Goat
662. Goat
663. Goat
664. Goat
665. Car
666. Goat
667. Goat
668. Goat
669. Goat
670. Car
671. Goat
672. Goat
673. Goat
674. Goat
675. Goat
676. Goat
677. Goat
678. Goat
679. Goat
680. Goat
681. Goat
682. Goat
683. Goat
684. Car
685. Car
686. Car
687. Goat
688. Car
689. Goat
690. Goat
691. Goat
692. Goat
693. Car
694. Goat
695. Goat
696. Goat
697. Goat
698. Car
699. Goat
700. Goat
701. Goat
702. Goat
703. Goat
704. Goat
705. Goat
706. Goat
707. Car
708. Goat
709. Car
710. Car
711. Car
712. Car
713. Goat
714. Car
715. Goat
716. Car
717. Goat
718. Goat
719. Goat
720. Goat
721. Goat
722. Goat
723. Goat
724. Car
725. Goat
726. Goat
727. Goat
728. Goat
729. Goat
730. Car
731. Goat
732. Car
733. Car
734. Goat
735. Car
736. Goat
737. Goat
738. Goat
739. Goat
740. Car
741. Car
742. Goat
743. Goat
744. Car
745. Car
746. Goat
747. Goat
748. Car
749. Car
750. Goat
751. Car
752. Goat
753. Car
754. Goat
755. Car
756. Goat
757. Goat
758. Car
759. Goat
760. Goat
761. Car
762. Goat
763. Goat
764. Goat
765. Goat
766. Goat
767. Car
768. Car
769. Goat
770. Goat
771. Goat
772. Car
773. Goat
774. Goat
775. Goat
776. Goat
777. Goat
778. Car
779. Goat
780. Goat
781. Goat
782. Goat
783. Goat
784. Goat
785. Car
786. Car
787. Car
788. Goat
789. Goat
790. Goat
791. Car
792. Goat
793. Goat
794. Car
795. Car
796. Goat
797. Goat
798. Goat
799. Car
800. Car
801. Goat
802. Car
803. Goat
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. Car
818. Goat
819. Car
820. Goat
821. Goat
822. Car
823. Car
824. Goat
825. Car
826. Car
827. Goat
828. Goat
829. Car
830. Goat
831. Car
832. Car
833. Car
834. Car
835. Car
836. Car
837. Goat
838. Car
839. Goat
840. Goat
841. Car
842. Car
843. Goat
844. Goat
845. Car
846. Goat
847. Goat
848. Car
849. Goat
850. Car
851. Goat
852. Goat
853. Goat
854. Goat
855. Goat
856. Car
857. Car
858. Goat
859. Car
860. Goat
861. Goat
862. Goat
863. Goat
864. Goat
865. Car
866. Goat
867. Goat
868. Goat
869. Goat
870. Car
871. Car
872. Goat
873. Goat
874. Goat
875. Goat
876. Car
877. Goat
878. Goat
879. Goat
880. Car
881. Goat
882. Goat
883. Goat
884. Car
885. Goat
886. Goat
887. Goat
888. Goat
889. Goat
890. Car
891. Goat
892. Goat
893. Goat
894. Car
895. Goat
896. Goat
897. Car
898. Car
899. Goat
900. Goat
901. Goat
902. Goat
903. Car
904. Goat
905. Goat
906. Goat
907. Goat
908. Goat
909. Goat
910. Goat
911. Goat
912. Car
913. Car
914. Goat
915. Goat
916. Car
917. Goat
918. Goat
919. Goat
920. Car
921. Goat
922. Goat
923. Car
924. Car
925. Goat
926. Car
927. Goat
928. Goat
929. Goat
930. Car
931. Goat
932. Goat
933. Car
934. Goat
935. Goat
936. Car
937. Goat
938. Goat
939. Car
940. Goat
941. Goat
942. Goat
943. Car
944. Goat
945. Car
946. Car
947. Goat
948. Goat
949. Car
950. Goat
951. Car
952. Goat
953. Car
954. Goat
955. Goat
956. Car
957. Goat
958. Car
959. Car
960. Car
961. Goat
962. Car
963. Goat
964. Goat
965. Goat
966. Goat
967. Goat
968. Goat
969. Goat
970. Goat
971. Goat
972. Goat
973. Goat
974. Goat
975. Goat
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Goat
982. Car
983. Goat
984. Goat
985. Car
986. Car
987. Goat
988. Car
989. Car
990. Goat
991. Goat
992. Goat
993. Goat
994. Goat
995. Goat
996. Goat
997. Goat
998. Car
999. Car
1000. Car

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 99 times and lost 901 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;
	}
}