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 5 times and lost 5 times.

1. Car
2. Goat
3. Goat
4. Car
5. Goat
6. Car
7. Car
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 322 times and lost 678 times.

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