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. Car
4. Goat
5. Car
6. Goat
7. Goat
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 337 times and lost 663 times.

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