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 5 times and lost 5 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Car
6. Car
7. Goat
8. Goat
9. Car
10. Car

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 321 times and lost 679 times.

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