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 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
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 332 times and lost 668 times.

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