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 4 times and lost 6 times.

1. Car
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Car
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 341 times and lost 659 times.

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