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 3 times and lost 7 times.

1. Car
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
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 337 times and lost 663 times.

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