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. Goat
6. Car
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 334 times and lost 666 times.

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