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. Car
4. Goat
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 335 times and lost 665 times.

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