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 4 times and lost 6 times.

1. Goat
2. Goat
3. Car
4. Car
5. Goat
6. Goat
7. Car
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 308 times and lost 692 times.

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