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. Car
2. Goat
3. Goat
4. Goat
5. Goat
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 336 times and lost 664 times.

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